Stat Assistant Professor

If you have a strong publication record and strong letters, this should not be an issue. For example, Yves Atchades (a professor at University of Michigan for awhile before recently moving to BU) has a PhD from University of Montreal.
You may need to do a postdoc, but even the majority of graduates from the top programs in the U.S. (who pursue academic careers) end up doing postdocs these days. 

Lol, $3000 for a consultant? I would not listen to this bad advice, OP. Most PhD programs in Statistics in the USA are swamped with international applicants. The top tier programs like Penn, Stanford, and Harvard can take their pick and typically do not consider any applicants who are not from the top 5% of their graduating class at ISI, Peking, Tsinghua, etc. The OP sounds like a strong candidate but the reality for international applicants is they need to be the top students from the most elite schools in their home country in order to have a reasonable shot at these schools (even if they get a Masters degree from a school like Duke in the U.S., that is still not likely enough to overcome the undergrad alma mater issue because graduate school grades are usually more lenient).
We provided the OP with a realistic range of schools they should apply to and even encouraged them to try a few "reach" schools like NCSU (Harvard and Penn Stats would almost certainly be a waste of money though).

I think your list of schools sounds very reasonable, and I think you should be able to get into some of those Biostat programs, particularly the lower-ranked ones who will highly value your research and the fact that you submitted three methodological papers to reasonably good journals. I think the B+ you received in Linear Models I is partly offset by your A in Linear Models II.
It may be worth applying to UW Biostat for good measure.

I think your list of schools sounds very reasonable, and I think you should be able to get into some of those Biostat programs, particularly the lower-ranked ones who will highly value your research and the fact that you submitted three methodological papers to reasonably good journals. I think the B+ you received in Linear Models I is partly offset by your A in Linear Models II.
It may be worth applying to UW Biostat for good measure.

Re: your chances at top-tier programs. Your profile looks quite strong, and I think you have a definite chance at the top Stats programs in the U.S. If you went to a top university in Australia like Australian National University, University of Sydney, or University of Melbourne, you should be fine. These schools are regularly ranked among the top 50 universities in the world, and they are also known for rigor in their mathematics programs. I have seen Australians from the aforementioned universities graduate from the likes of Stanford Statistics and UPenn Wharton Statistics (e.g. Giles Hooker). For someone like you, I would recommend applying to most/all of the top programs in the U.S., including UC Berkeley, UChicago, Columbia, Washington, etc. and a few large state schools like Texas A&M, Minnesota, and Purdue for good measure. I think you should definitely be able to get into University of Washington, and you have an excellent shot at a place like Berkeley as well. 
Re: letters of recommendation. You should aim to get the strongest possible letters of recommendation from people who can speak to your research potential. You do not need to have published any papers to have a chance at Statistics or Math PhD programs (that's what the PhD program is to help train you to do!), but you need to display evidence of research potential in your application. I think you have pretty good evidence of that. Your supervisor who is the editor of an academic journal would be fine as an LOR writer, as would the supervisor in applied health research. Ideally they would mention in their LORs that they work in a research capacity and that they conduct research that is published in academic journals. At least one letter should also point to your excellent performance in advanced mathematics and your strong mathematical aptitude.
Re: subject GRE. You will need to take the math subject GRE if you want to have a chance at Stanford. Otherwise, you could probably do without it.
Re: your research interests. Off the top of my head, University of Washington is strong for TDA, and Carnegie Mellon is strong for both TDA and differential privacy. I would say that for your specific interests, UW and CMU are two of the strongest programs in those areas among the top-tier programs in the U.S. (though it's possible that other top-tier programs are also particularly strong in these two areas, I just may not be aware of it). 

Most Mathematics MS programs allow you to do a concentration too. I would suggest the OP do a mathematics Masters degree with a concentration in statistics. This way, he can take the Casella and Berger mathematical statistics sequence, while also taking the usual real analysis, abstract algebra, and topology classes. Getting A's in all those would help mitigate the B's he previously received and show that he can do proof-based work.

Absolutely. I came from a combined math/stats department, and the difference in TA appointments was great. For incoming masters students, there would be around 10-15 appoints for math students, and 1-3 for statistics.

The most important parts of the faculty application are the letters of recommendation and your publication record. If you have a strong publication record and can secure great letters from famous people (especially from people who are not your PhD/postdoc advisor or members of your thesis committee -- that actually helps a lot since it means your work is getting noticed by people who don't have a personal stake in your success), you should be in a good position to find a faculty job.
Prestige of PhD granting institution is certainly correlated with both of the above (partly because the top programs have more famous professors who are more prolific at publishing in the top journals), but a PhD from a mid-tier or lower ranked school seems to be much less of a hindrance in STEM than it is in humanities/social sciences. For example, I have seen faculty at Harvard University (including Assistant Professors and Department Chairs) in computer science, biostatistics, and physics whose PhDs are from UMass Amherst, University of Florida, and UC Boulder. The Associate Chair of the Harvard Biostats Dept did his PhD at University of Florida.
My advice is to do good work, secure a good postdoc, and network with the top people in your sub-area. The rest will follow.

@playingstats Thanks for the clarification. As others have mentioned, your biggest hurdle for PhD admissions will be convincing adcoms that you can handle the rigorous mathematical coursework. Apart from possibly some applied regression, categorical analysis, and statistical computing classes, the majority of classes you take in your first two years of a Statistics PhD program will tend to be very mathematical and proof-intensive (particularly classes like advanced statistical inference, measure-theoretic probability theory, theory of linear models, and large sample theory ). Later when you do your dissertation research, you can choose to focus on something more methodological/applied, but to get to that point, you have to pass written qualifying exams on proof-intensive courses. And even if your research is more applied, you still need to have advanced knowledge of the math behind it (and some basic knowledge of theory is also helpful). Just as an example, if your research is on Bayesian nonparametrics (say), you could write applied papers that use a Dirichlet process (DP) prior without any theorems/proofs, but you still need to understand the mechanisms that make the DP prior work.
Since you are currently employed by a university, I assume that they give you some sort of benefits to take courses there at a reduced tuition? In order to strengthen PhD applications down the road, you could take a few advanced undergraduate math classes and get A's in them. For example,  you could retake Real Analysis and take an Advanced Linear Algebra (with proofs) class. If you can get A's in these courses, that would greatly assuage Statistics adcom's concerns that you cannot handle the math, particularly the mathematical proofs component. Masters degrees at reputable universities are also something to look into, and strong performance there would certainly help your profile a lot. Since you are a veteran, you may be able to get some educational benefits and not go into a ton of debt to obtain a Masters degree. If you are unable to retake Analysis or take advanced LA now, you should definitely retake Real Analysis in a Masters program (it can be real analysis at the undergrad level, it does not have to be the PhD-level measure theory class) and maybe a few other upper division math classes. As the others above me mentioned, you need to get A's in these courses to have a chance at some Statistics PhD program.
In your case, a Masters in *Mathematics* may actually boost your profile more than a Masters in Statistics for admissions to Statistics PhD programs, because this would prove that you can get A's in math classes and partly mitigate the B's from your undergrad. And Mathematics MS programs are more likely to be funded than Stat ones, since they often need TA's for  large college algebra/trig, pre-calc, and Calculus courses. So this is something to also consider.

@omicrontrabb nails it.
To the OP (and other prospective applicants): It is very important for prospective Statistics PhD applicants to know that Statistics PhD programs (much like math and computer science PhD programs) are designed to train you more in the spirit of a mathematician/scientist, not as an analyst or programmer who implements ML algorithms and does routine data analysis. If doing mathematical proofs and designing new methodologies and algorithms that require heavy knowledge of the underlying mathematics are not things that sound appealing to you, then I would recommend just getting a Masters. Just as you do not need a PhD in CS to become a software engineer, you don't really need a PhD in Stats to work in data science (with a few rare exceptions like research scientist positions at places like Microsoft Research, Google.ai, etc.  and obviously, for international students, it is easier to obtain run-of-the-mill data science jobs with a STEM PhD, but this shouldn't be an issue for you).

@playingstats Thanks for the clarification. As others have mentioned, your biggest hurdle for PhD admissions will be convincing adcoms that you can handle the rigorous mathematical coursework. Apart from possibly some applied regression, categorical analysis, and statistical computing classes, the majority of classes you take in your first two years of a Statistics PhD program will tend to be very mathematical and proof-intensive (particularly classes like advanced statistical inference, measure-theoretic probability theory, theory of linear models, and large sample theory ). Later when you do your dissertation research, you can choose to focus on something more methodological/applied, but to get to that point, you have to pass written qualifying exams on proof-intensive courses. And even if your research is more applied, you still need to have advanced knowledge of the math behind it (and some basic knowledge of theory is also helpful). Just as an example, if your research is on Bayesian nonparametrics (say), you could write applied papers that use a Dirichlet process (DP) prior without any theorems/proofs, but you still need to understand the mechanisms that make the DP prior work.
Since you are currently employed by a university, I assume that they give you some sort of benefits to take courses there at a reduced tuition? In order to strengthen PhD applications down the road, you could take a few advanced undergraduate math classes and get A's in them. For example,  you could retake Real Analysis and take an Advanced Linear Algebra (with proofs) class. If you can get A's in these courses, that would greatly assuage Statistics adcom's concerns that you cannot handle the math, particularly the mathematical proofs component. Masters degrees at reputable universities are also something to look into, and strong performance there would certainly help your profile a lot. Since you are a veteran, you may be able to get some educational benefits and not go into a ton of debt to obtain a Masters degree. If you are unable to retake Analysis or take advanced LA now, you should definitely retake Real Analysis in a Masters program (it can be real analysis at the undergrad level, it does not have to be the PhD-level measure theory class) and maybe a few other upper division math classes. As the others above me mentioned, you need to get A's in these courses to have a chance at some Statistics PhD program.
In your case, a Masters in *Mathematics* may actually boost your profile more than a Masters in Statistics for admissions to Statistics PhD programs, because this would prove that you can get A's in math classes and partly mitigate the B's from your undergrad. And Mathematics MS programs are more likely to be funded than Stat ones, since they often need TA's for  large college algebra/trig, pre-calc, and Calculus courses. So this is something to also consider.

@playingstats Thanks for the clarification. As others have mentioned, your biggest hurdle for PhD admissions will be convincing adcoms that you can handle the rigorous mathematical coursework. Apart from possibly some applied regression, categorical analysis, and statistical computing classes, the majority of classes you take in your first two years of a Statistics PhD program will tend to be very mathematical and proof-intensive (particularly classes like advanced statistical inference, measure-theoretic probability theory, theory of linear models, and large sample theory ). Later when you do your dissertation research, you can choose to focus on something more methodological/applied, but to get to that point, you have to pass written qualifying exams on proof-intensive courses. And even if your research is more applied, you still need to have advanced knowledge of the math behind it (and some basic knowledge of theory is also helpful). Just as an example, if your research is on Bayesian nonparametrics (say), you could write applied papers that use a Dirichlet process (DP) prior without any theorems/proofs, but you still need to understand the mechanisms that make the DP prior work.
Since you are currently employed by a university, I assume that they give you some sort of benefits to take courses there at a reduced tuition? In order to strengthen PhD applications down the road, you could take a few advanced undergraduate math classes and get A's in them. For example,  you could retake Real Analysis and take an Advanced Linear Algebra (with proofs) class. If you can get A's in these courses, that would greatly assuage Statistics adcom's concerns that you cannot handle the math, particularly the mathematical proofs component. Masters degrees at reputable universities are also something to look into, and strong performance there would certainly help your profile a lot. Since you are a veteran, you may be able to get some educational benefits and not go into a ton of debt to obtain a Masters degree. If you are unable to retake Analysis or take advanced LA now, you should definitely retake Real Analysis in a Masters program (it can be real analysis at the undergrad level, it does not have to be the PhD-level measure theory class) and maybe a few other upper division math classes. As the others above me mentioned, you need to get A's in these courses to have a chance at some Statistics PhD program.
In your case, a Masters in *Mathematics* may actually boost your profile more than a Masters in Statistics for admissions to Statistics PhD programs, because this would prove that you can get A's in math classes and partly mitigate the B's from your undergrad. And Mathematics MS programs are more likely to be funded than Stat ones, since they often need TA's for  large college algebra/trig, pre-calc, and Calculus courses. So this is something to also consider.

Sounds like you have a better understanding of what a PhD would be like than half the people I know who started one. I agree with the above posters that you're not going to get into any of those schools (besides perhaps Bowling Green, and maybe an unfunded offer from FSU).  Realistically, I don't see you getting into a program ranked above 60 right now. If you want to apply to PhD programs, I would start around there. Of you don't want to be a professor at a PhD-granting school, it matters much less where your degree is from, especially for government/most of industry. If you would like to aim higher in a couple years, I think you could likely get into a decent MS program, including at some of those schools you mentioned (although I really don't know much about MS admissions). In order to improve your profile, make sure to go somewhere where you can take mathematically rigorous classes, get As in them, get to know some accomplished professors to get letters from, and perhaps do some research, and you can improve those results. 

I agree with @bayessays  You mention that you would like to ultimately become a professor. 
Your most realistic path to that is probably something like this:
Getting a masters degree. You could probably get into somewhere around the level of FSU. According to the FSU website, their average admitted GPA was around 3.3. You're probably not going to get funding. Do well in your masters. Get As in mathematically rigorous classes and get some research experience. Some programs allow you to transfer from the MS to PhD or apply to other programs around that level. If you look at the placements from FSU, almost all of their graduates go into industry. That might be selection bias since they can earn far more in industry. The ones who go into academia generally go to schools ranked around FSU or lower.
I don't think you have a realistic chance at places like Texas A&M or Purdue even with a masters degree.
I'm not very familiar with masters admissions, so other people might have better advice than me.

@omicrontrabb nails it.
To the OP (and other prospective applicants): It is very important for prospective Statistics PhD applicants to know that Statistics PhD programs (much like math and computer science PhD programs) are designed to train you more in the spirit of a mathematician/scientist, not as an analyst or programmer who implements ML algorithms and does routine data analysis. If doing mathematical proofs and designing new methodologies and algorithms that require heavy knowledge of the underlying mathematics are not things that sound appealing to you, then I would recommend just getting a Masters. Just as you do not need a PhD in CS to become a software engineer, you don't really need a PhD in Stats to work in data science (with a few rare exceptions like research scientist positions at places like Microsoft Research, Google.ai, etc.  and obviously, for international students, it is easier to obtain run-of-the-mill data science jobs with a STEM PhD, but this shouldn't be an issue for you).

Realistically, you're not going to be admitted to a top 50 statistics PhD program with a 3.3 GPA from an unknown school. Your coursework at Harvard Extension is unlikely to help. Most students get As in those sort of programs. An admissions committee will be seriously concerned about your ability to do the advanced math required for a stats PhD since you have Bs in linear algebra, real analysis and numerical analysis.
I attended the visit day for Ohio State's stats PhD program this year. This seemed to be a typical admitted student:
Attended a well known, but not prestigious university. Think something like UConn. Math or stats major with mostly As. Most had math or stats research experience. I assume the admitted students at the other programs you listed would be broadly similar.
What is your goal? What are you hoping to gain from getting a PhD in statistics?

For Masters in Statistics/Biostatistics, you mainly need to have good grades in Calc I-III and linear algebra and a solid GRE Quantitative score. Other stuff, including relevant internships, research, etc., doesn't matter that much. I think with your current profile (and contingent on a good GRE Q score), you should have no difficulty getting into the vast majority of Statistics Masters programs.
I think picking up a second major in mathematics would be highly beneficial in case you want to apply to PhD programs down the road. It seems you should be able to handle this, given your current profile.

Your profile is outstanding. UCLA has one of the best math departments in the world (#7 in USNWR), and your grades and test scores are outstanding.
The lack of research experience should not pose an issue, but just one question: you say that you are getting two letters of recommendation from "lecturers." Are these not tenured professors or tenure-track Assistant Professors? It would be far more preferable for you to get strong letters from tenured or TT professors (not lecturers or adjuncts) who can credibly speak to your research potential. Also, you do not need to submit more than three LORs, and the adcoms may only look at three of them (given time constraints), so I would recommend submitting three of the strongest ones you can from professors. They do not need to come from stats professors; they could all be from math profs.
Anyway, if you apply to all the top 10 programs AND get strong letters of recommendation, you will probably get into at least a few of them.

The SOP doesn't have to be amazing, just competent (i.e. you can detail your academic and research experience and state a few potential research areas of interest). The letters of recommendation, grades (especially those in upper division math classes), and the general GRE scores are the most important parts of the application. The GRE is mainly a filter though, but a high score on it can sometimes lead to on-campus fellowships. Your GRE scores are exceptional -- even better than a lot of domestic applicants (the GRE V score is certainly very impressive for someone whose first language is not English).

Given that you are from one of the top 3 schools in South Korea and your overall grades in math/stat classes are pretty good, your list of schools is very reasonable. You do have a few B's, but most of your other grades are excellent. I don't think you need to worry so much about the fact that you switched from business to stats. I know people from South Korea whose undergrad major was business (they only took math courses in their last few years so they could apply to Stats graduate programs in the U.S.)
UNC-STOR may be hard to crack, though. This department is tougher to get into than the others you listed, and furthermore, it is also more probability theory-focused than others. I would recommend you apply to NCSU.

Given that you are from one of the top 3 schools in South Korea and your overall grades in math/stat classes are pretty good, your list of schools is very reasonable. You do have a few B's, but most of your other grades are excellent. I don't think you need to worry so much about the fact that you switched from business to stats. I know people from South Korea whose undergrad major was business (they only took math courses in their last few years so they could apply to Stats graduate programs in the U.S.)
UNC-STOR may be hard to crack, though. This department is tougher to get into than the others you listed, and furthermore, it is also more probability theory-focused than others. I would recommend you apply to NCSU.

For Masters in Statistics/Biostatistics, you mainly need to have good grades in Calc I-III and linear algebra and a solid GRE Quantitative score. Other stuff, including relevant internships, research, etc., doesn't matter that much. I think with your current profile (and contingent on a good GRE Q score), you should have no difficulty getting into the vast majority of Statistics Masters programs.
I think picking up a second major in mathematics would be highly beneficial in case you want to apply to PhD programs down the road. It seems you should be able to handle this, given your current profile.

Casella and Berger is the core of your first year of courses in a MS/PhD program and will be theory-based.   I'd recommend you take that because it will 100% help you in a PhD program. 
Machine learning is likely to be more of a survey course with some applications and coding. It might help you become familiar with some common methods and get some experience coding, but it's unlikely to provide that base of knowledge that will help you in a PhD program.  On the other hand, it is always helpful to know the ML buzzwords if you want an industry data science job.  I highly doubt an undergraduate ML course will go into much depth of the theory.

It could possibly help your application to ask one of your LOR writers to mention that you plan to take certain advanced math classes in the spring semester of your senior year (like Real Analysis II and whatever other advanced math/stat classes) to ensure that you are prepared for graduate school. But the most important things are the grades on the transcript you submit with your application (which won't have your spring 2020 grades) and the letters of recommendation.
I think you should do well in admissions for most of the schools ranged 16-40 and your chances are above average at Duke and UW (conditional on good performance on the GRE and in Real Analysis and Advanced LA). Penn State has a good reputation, and physics majors who have also taken some advanced math (namely real analysis) are usually looked favorably upon by Statistics adcoms. In fact, an alumnus who just graduated from my PhD alma mater got a TT job at University of Minnesota Statistics (without a postdoc) and his undergrad major was Physics.

Assuming that you score well on the Quantitative section of the GRE and get A's in Real Analysis I and Advanced Linear Algebra, I think you have a definite shot at NC State and possibly Duke. Conditional on strong performance there, I think you could even try applying to a school like University of Washington (though this is possibly a reach). Physics is a hard subject, and your GPA is pretty good. 
I'd recommend adding a few more schools like Wisconsin or Minnesota too.