rockwizard

Regarding the Quant Section of the GRE, I believe most of the mathematical concepts that require mastery in order to obtain higher scores become pretty useless to the vast majority of programs once you hit the 50th percentile. I also think that this section favors "traditional" students over those who have taken time off to research, travel, work, etc. since they will have most of these mathematical concepts fresh in their minds (thus, a bit of ageism might be at play). My thinking is that this portion of the test is really more of a test of a student's level of discipline, since a PhD program's curriculum will likely cover most of the statistical methodology that would be considered of value to the research program, and many young researchers will already have exposure to this through publication experience and undergraduate coursework. It's a bit strange to me that the GRE Quant section does not cover any statistical concepts, as I would think these would encompass the most wide ranging applicability (if you truly wanted to accomplish the impossible and apply a "one-size-fits-all" test to every graduate academic program). I'm not sure if theoretical scientists would feel singled out by this since I am not quite familiar with their lack of experimental design, but who cares about them since they are a bunch of weirdos who tend to transcend the status quo anyways.
Concerning the Verbal and Writing portions of the test, research academia does not really require eloquent writing abilities in order to publish significant findings (unless you're writing a review article, maybe?) and some could even say that verbal ability as a factor in publication potential encourages ethnocentric bias in a multicultural academic environment. However, realistically speaking I would probably want one of my students to be able to construct a solid argument without me having to revise their work extensively (although I might also consider this the "fun part" of the publication process since I actually enjoy writing). Again, I might say a good cut-off here is at about the 50th percentile.
I definitely think the GRE can currently help sort out the "numbnuts" factor in a potential PhD candidate, but it's also a bit like asking someone to practice quilting in preparation for a martial arts tournament. In my opinion it is utterly boring and very likely useless. To be fair, I suppose a high score does demonstrate a level of commitment and discipline. However, the reason I am attempting to enter this field is because of my fascination with the subject matter and I would rather spend my free time (which is pretty scarce) on research projects instead of jumping through hoops. Academia probably needs to evolve at some point and adapt different tests that focus on each specific field that fuse quantitative, verbal, and subject-specific based content in a way that is appropriate to every program. That argument might essentially consist of trying to convince ETS to make less money.
Anyhow, if you have been screened by a program because of the GRE (and are stubborn about not re-taking it), maybe you should instead focus on writing an incredible grant/fellowship proposal to excel in a different program or bring it back to the table during the next application cycle. The "brand name" of a school can only go so far, and at some point you will have to realize that the ability to coalesce your expertise, interests, and initiative into significant scientific discovery is far more valuable than just hopping aboard the reputation train of someone else's program. If you manage to excel in this, you can probably wind up directing your own research in a faculty position at one of the institutions that screened you for some stupid test with subject matter that possesses little relevance to your career path. Maybe you'll even help form a prominent new research institute :).

Yeah, honestly if a professor is the latter it's probably a sign I don't want to work for them anyway. I've heard from many professors that they don't even glance at GRE scores so. 
Yeah, I'm like this too. Which is, in part, why I ran out of time on the GRE. The whole "being treated like I was in a high security prison" thing during the test didn't help with my anxiety either.

I'm not sure what type of pride this is, but your applications indicate otherwise. If you wanted to get into a mediocre school you would have applied elsewhere. 
So you didn't prepare for it much. Okay that's fine, prepare for it and you should do much better.  
Also the GRE doesn't test any college math, it tests math up to college level as it's made for students from engineer all the way to psych/socio who don't have to take anything beyond precalc. It's a test of logic problems, if you're sitting down to physically write out much math you're doing it wrong.  Based on this it seems like you fell for the trap the GRE is supposed to catch people in. 
I could sit here and enthrall you with a history of the socioeconomic status of my family to pity justify myself to you but I won't.  Nothing I said was intended as a derogatory remark, but I'm sorry I may have wrote it in a way that misconstrued it to you as such; additionally dirt poor is an expression in the rural midwest where I'm from.  Anyway you've spent what $50000-150,000+ on an education over 4+ years of life? The GRE's 200 dollar cost (100 with the low-income voucher if you qualify) and a month of study. It's a a small sum by comparison, find a way, I know poorer people than you who have.
https://pdfs.semanticscholar.org/5c69/875be977c1de9392dd8392937ebbf606dd18.pdf
This is a very well cited meta-analysis of the GRE and its predictive value for graduate student success and correlates that a good GRE corresponds to success in graduate school.
If you reject this fine, but for the time being the GRE is used by everybody and the system isn't going to change soon.  Don't put yourself out of it due to a stupid test.
 
Anyway, take my advice or don't, but don't try to bite a hand that's trying to do nothing but help you succeed. 

The GRE isn't a great predictor for students who all range around the same (e.g. an 88th vs a 74th is likely to be irrelevant).  However, for students who range outside of the traditional range or who have a great package but a bad GRE it's a red flag.  In fact probably one of the biggest, as the GRE is the only thing the student has complete control over (even grades are subjective sometimes with certain institutions giving As out like candy).  This comes from a variety of admin councils I've spoken to, discussions between faculty and students at seminars I've organized, and so forth.  I agree the GRE is a roadblock for low-income students, but in reality being low income is a roadblock in life in general, the GRE isn't going to be the ultimate roadblock here.  Additionally, no matter how sad your story in a grad application there's always somebody with a sadder one. 
Harvard EPS has a requirement of the GRE but allows you to explain why you didn't take it if you don't submit the scores.  Not really truly 'optional.' If anything that seems more of a test for everybody who isn't dirt poor or from some very impoverished nation with no infrastructure. 
Funding is an X factor we can't control for, so I don't even both.  However it's unlikely that these 3 schools, all of which have considerable financial resources, couldn't scrap together a TA-ship to support a stellar candidate for a year while a POI gets money together. 
You are correct with the SOP.  By far the most important piece of a application, but I defer to the boat analogy.  

To Rockwizard (sorry unsure how to tag people here, or if that's a thing).  Allow me via the anonymity of the internet to have a frank conversation with you.
You're background in research is stellar.  GPA is a bit low, but nothing bad and as long as it wasn't showing a downward trend in your junior-senior year will be washed out by your research easily. 
That said CU and Harvard (probably Arizona) will look for a few type of candidates coming form a pure undergrad background, these are generalizations but you'll get the point:
Candidate A:  High GPA, High GRE - these are your 'that guy/girl' valedictorian students.  What filters these people out is how much they bank on their high scores to get in.  If they're just in it thinking this is high school and good grades get you everywhere they're likely to be rejected.  They will need a good research focus to be let in, even with top marks. Some risk with these students, top marks doesn't mean good researcher. 
Candidate B: High GPA, High GRE (probably less so than Canid A, but still high), and research experience - these are you admits everywhere. Proven proficient students and proven proficient researchers. The total package.  Almost no risk in them failing.
Candidate C Okay GRE, Okay GPA, and research experience - these are admitted some places, rejected others.  Entirely based on the discretion of the POI and admin council.  Usually what gets these people in is research experience or a SOP that's focused towards a particular avenue of research at that school indicating they'd be a good fit. For example: this candidate might like seismic geophysics and have some background in it.  In their SOP they indicate the seismic geophysics people at that school and discuss briefly how they'd really like to be involved in their work. 
 You're a candidate C, or you could be as your GRE is an issue.  151 on quant is ~43rd percentile or so.  CU and Harvard will see candidates with stellar GREs and GPAs applying,  and almost all will filter out anything below a 50th, and most schools will filter out anything below a 70th (even though they say they wont).  Now this isn't to say you need a 95th+ on everything to beat out all of those 70th percentile cutoff people, you don't, research is far more important than GPA and GRE and you've demonstrated you're proficient in that.  WHAT YOUR GRE IS HOWEVER is this analogy:
You've built a really solid ship, great potential, this thing could sail the oceans for decades, and it's got all the amenities I'm looking for, nice cabin, a pool, open bar ------ this is your research experiences and background
But prior to launching this ship you decided to torpedo the side of it ------ this is your quantitative GRE. 
Now that ship could still float, heck it could make it across the oceans forever.  But if I'm left with the options between that torpedo-ship or another ship that doesn't have a torpedo hole, I'm going to go with the non-torpedo ship.
Get the quant up, it's dragging you down. Consider doing research for a year while you stage for 2020 applications. 
Let me stress this to you:
YOU HAVE A HARVARD, MIT, COLUMBIA, name a fancy school and it fits, application package in the making. But you've shot yourself with that GRE. 
Your PhD is merely delay, not on hold, and it's only delayed a year.  Spend the hundreds of dollars again to retake the GRE, it's trivial compared to how much you've invested in your education already. 
That said I hope you get into Harvard and this conversation is moot. 

Maybe -- but for example, Harvard EPS has moved to GRE optional stance to my knowledge because of how standardized tests add cost to applicants and only vouch for things like socioeconomic status, both of which reduce diversity. As someone who got into Harvard with a terrible undergrad record, I'm much more convinced it's the statement that makes the biggest difference in any package. If you have a clear goal for what you want to do, how your trajectory has prepared you for that work, and how this slots into your present view of your long-term aims, you're okay. Add in what benefit the department will get from you being there (as opposed to just what you will get out of the relationship) and you're even better. It also never gets said enough about how things like funding make a huge difference...the same application in 2 different years could see 2 very different results. Same goes for advisors, one might be okay taking a chance on someone, others (I can think of one in my current dept) are much more particular about boxes applicants have to check to be considered qualified for entry, but as long as your POI is vested enough to pull you in, that's all that matters. Of course, none of us have sat on adcoms and so this is mostly speculation, but this is my 4th application cycle so I like to think I've gotten some anecdotal sense of what can help you rise in the pile.

Not industry, but IT and web development. My first job after graduating was half support for Linux servers, half web development. Then I found a software engineering job. I had done a lot with Linux, MATLAB, and python for a few semesters and a summer REU, which helped me convince people to pay me to do this stuff. At the time, I didn't want to move, so my job opportunities were limited to what was local. 
Good luck with Harvard!