Buddenbrooks

In this case I would definitely apply to Chicago and Columbia. Although they are tough to get in, I think your chances are above average.

Having those types of connections may indeed be helpful. It could also depend on who is on the admissions committee.
It just seems as though most of the "top" Statistics PhD programs are extremely selective about what domestic applicants they admit (to the point that some schools like Harvard and UPenn only have 1-3 domestic students enrolling each year). Some top programs like UC Berkeley, University of Washington, and Duke are a bit more flexible (in that they admit more Americans and not all from very prestigious undergrads).

I still think the likes of Columbia and Chicago might be tough. There was a combined Bachelor's/Master's American student at the school where I got my PhD from (a large flagship state university) who had done legitimate research on theoretical probability, had taken basically the entire Statistics PhD curriculum *and* a bunch of math PhD classes (measure theory, functional analysis, etc.), and he was still rejected from most of the top statistics PhD programs (he did end up at a very good program though -- one of the four "reaches" mentioned by the OP). I think the top schools are very competitive, and a sub-3.8 undergrad GPA from a less prestigious uni may have a more difficult time cracking schools like Columbia or Chicago. These schools don't accept many domestic students to begin with.
That said, the OP does have a very good profile, and I would encourage them to apply to more schools in the top 20 if they can afford it. 

The overall GPA usually serves as a cutoff and you can definitely pass the cutoff so I wouldn't worry too much about it. Since you have strong background in math did well in all upper division courses, I think your profile will be viewed more favourably than someone with 3.9 GPA and low grades in real analysis. I would apply to 10-20 schools since admissions are much more competitive than before. I agree that Chicago/Columbia might be reaches but I think you have a good chance of getting into UNC/Wisconsin/Penn State. 

You have very strong mathematical background. A good subject test score will definitely help at top places like Washington/Columbia/Chicago and I think a 80% or even 90% is attainable with some preparation given your background. I think you have a good chance of getting into top 20 and you don't need to go lower than UCLA/Rice.

Your profile looks pretty strong. I think Wisconsin, Penn State, UCLA, and Rice are definitely possible. Columbia, Michigan, UNC, and Washington might be slight reaches, as these are very competitive programs and there will be other domestic applicants to these programs who have better undergrad GPAs from more prestigious schools. However, your research experience is very solid, and your graduate school performance should instill enough confidence in your math abilities.
If I were you, I would apply to 4 schools in the range of Wisconsin to Rice and then pick two schools from Columbia, Michigan, UNC, and Washington to apply to (of those, I think Washington and UNC are more likely to admit you).