The biggest factor I am trying to understand is what Yale's value added is (and why) in those job searches. Why does DOJ care if my diploma says YLS on it?
I say Yale in part because of the way PI hiring works: as I think you're aware at this point, it's nothing like biglaw hiring, and you basically have to bust your ass all the way through school to nail down a paying PI job at the end. (Unless you're at YLS.) If you go to Chicago, you may watch many of those internships, summer positions, clerkships and eventual jobs go to YLS students with similar or less experience than you.
Why? Because the profession you're entering is a prestige-obsessed one. It seems like that's the argument you're trying to have: Why does the legal profession care so much about the difference between Yale and Chicago? It's stupid and baseless! This is fine as an intellectual exercise, and if you go to Yale it'll be a good thing to keep in mind, but in terms of your decision-making you're not going to get the answer you want, which is: Go to Chicago! It won't matter!
It is stupid and (mostly) baseless, but this is the profession you're entering. It may not be fair that YLS grads would generally be privileged over a similarly qualified Chicago candidate, but it's reality. You will likely make life a lot easier on yourself in terms of the hiring process and the stress you'll be under if you pick Yale now. You may literally save yourself two or three years of work (or more) to get where you want to go by getting a YLS degree, plus a hell of a lot of stress 2L and 3L years (plus other intangible benefits throughout your career that I won't emphasize too much for fear of getting the TLS Debt-Averse Hammer thrown down upon me). Is that worth it for the debt? With COAP, I'd say it's worth it, no question.
In terms of alumni, the best-represented schools in prestigious civil rights impact lit are probably Yale and NYU. If you were offered a full-tuition scholly + living at NYU, that'd be an arguable choice given your career goals and NYU's great PI rep, but NYU doesn't have a Ruby equivalent, right? (NYU for PI is another thing that's generally lost on TLS.) Again, it's not that you can't get where you want to be from other schools, just that you're making the process of getting the job you want harder for yourself.
TLS gives great advice to the majority of 0Ls, but you're a unicorn-y type looking for a unicorn-y job, and this may not be the best place for you to seek advice. There's a very small handful of people on TLS who know anything about prestigious PI hiring or the PI job market. That gets pretty negligible when you look at civil rights lit specifically.
If you are still unsure (which you should be, given that I'm an anonymous stranger), I would rely on the advice of the practicing attorneys you spoke with, talk to a few (indebted, COAP-eligible) YLS alums 5 or 6 years out (and, if you can find them, Chicago grads practicing in your specific area of interest), and maybe discuss this with some PI-focused 2Ls and 3Ls who can help.