I don't know how much I have to help here -- I have never sat on an admissions committee! However, I have this to contribute: I'm an international student who has received offers this round from multiple universities on your list. (Yeah, things have gone pretty well for me.)
I have roughly your scores on GRE + mGRE. I do not have your background -- I could not pass Harvard quals right now. (I have a *strong* background but generally know 80% of the things they expect for each topic, but not the last 20%. One of them, though, I've never taken a proper course in, so that would be a fair bit of work.) Two things that I suspect helped: I have done original research in maths -- a year-long thesis, which didn't quite get me anything publishable but got very very close. (That's the plan for my new few months of spare time!) As a result, that letter of reference was, I think, exceptionally strong, and I linked my work wherever I could in the application. (My other two letters were strong -- one based on a research project, one on coursework by someone who's a bit of a mentor -- but probably not quite as good.) All of my letters were from people who did PhDs and postdocs in the US at strong schools.
My read on things: you've done enough research that you can say, with evidence, that you like it and can do it. Most US students, from my understanding, have never spent e.g. six weeks working on a particular question while making no progress. (Joy of joys. I've done this twice; the second one I actually figured it out! Still was a pretty frustrating six weeks.) You also have, as you say, a very strong background. All of this is great.
However, the selection process is pretty random. A good friend of mine, who imo is a stronger mathematician than I am, did not get into one of the top universities that I did. (They've got into some awesome ones, though, but just to say that there's a lot of variance.) While obviously things have gone well for me, the (US background) researchers I talked to were very strong on not relying on getting into a top uni for me. I am not Terry Tao, I am not a shoo-in, and at the end of the day, there are 400 people applying for 12 spots at a lot of these places. More than 12 of those are going to be excellent.
Definitely think about how your interests fit with the uni you're applying to. What areas of maths are you interested in? Are there people at those unis who do what you want to do? Some of the ones on your list are ones I did *not* apply to because there were not enough potential supervisors in my field. Also, consider applying to some unis one step down in prestige that still have the name recognition you want. You don't want to put all your eggs in this highly nerve-wracking basket. (I'm thinking UCLA, Yale, Brown, Cornell...) Good luck! Maybe I'll see you in the US in a couple of years