PhD Dilemma: Which program to choose?

If your goal is to get back into the quant space, you may know better than me, but my personal opinion is that it doesn't matter which of these schools you go to given that you already have experience working at a top fund. The difference in outcome that you would get from choosing one school over the other is very likely to be only worth an epsilon in that regard.

The only thing I could really see making a difference is if
1. There is a particular shop you're going for and you won't take anything else, then maybe you should see the backgrounds of people who work there and go to a school that most of them went to.
2. One school will afford you a research topic or courses that is more ML focused than the others (given the way the industry seems to be going). Frankly, however, I don't think this is that important since you can learn this anywhere even if your research isn't related to ML per se.

Now, the answer does become different if instead you were dead set on academia. In that case you would need to do some due diligence and check the alumni of these programs and see where they end up. Does everyone from the Stanford program go to industry? Then don't go there if you want to be a professor.

Similarly, if academia will be your goal you need to consider what topic you would study at each place and whether or not these topics are "hot". I've seen plenty of just "good but not great" students get TT positions at good schools simply because they studied an area that was "in", and I've seen tons of Math geniuses get low tier or only post-doc offers because they studied some esoteric area that no one else cares about.

Personally, i'm partial to Princeton, CMU, Stanford, but then again my PhD is in Statistics :). Yale is a good department for sure, but tbh I think it's the weakest one of all the offers you've gotten. All of those schools have some really heavy hitters that you could work with (for example Jianqing Fan at Princeton).
I think you're right that it doesn't really matter which program I choose as all of them are fine programs. I am probably overthinking this a bit.

Regarding the academia issue: I am indeed playing with the thought of staying in academia. Not super seriously, but I at least want to keep the door open. When looking at the placements, the Stanford program does not look as strong in my opinion. Many of the graduates became professors at business schools, which certainly is not my goal. Also not too many graduates ended up at tier 1 prop trading firms and hedge funds. For Princeton, both the academic and industry placements seem better ( I would indeed be very interested in working with Jianqing Fan. Rene Carmona's research on mean-field games is also something I am very interested in. However, I am not yet dead set on any specific topic and prefer to explore a bit in the beginning.

What's your take on Oxford? Do you see a difference between the US and UK programs?
First - I wouldn't discount business schools. Several of my classmates went on to positions at business schools (for example UPenn Stats is part of Wharton). The main benefit of being in the B school is that you get paid more than you would in the "pure" department. When you're a professor there's nothing stopping you from having collaborators (or even having a joint or affiliate position) with your "pure" department of interest, and the extra 50-100k you get from being a B school professor sure is nice.

Again, I don't think I'd worry about shop placements. Your experience at a top fund currently is going to get you interviews wherever you want, so school is probably not gonna make a difference there.

I can't comment on Oxford as I don't know much about it. Of course, opinion in the US is that it is an excellent (probably second only to Cambridge) school in the UK (and Europe at large). That said, if you look at the faculty pages for many departments I don't think you'll find many professors with PhD's from Oxford/Cambridge. I can't say why this is - it could just be that people in the UK or Europe prefer to stay there, so PhDs from Oxford/Cambridge just don't come to the US to be professors. However, if academia (especially in the US) is a possibility for you, I'd at least keep this fact in mind or try to figure out why it is.