I started out on Team Yale, and I am still on it. It still seems to fit the OP best overall.
But none of these schools constitutes a bad choice, so the OP really can’t make a poor decision as long as it is, in their estimation, the best fit.
Editing to add:
If you like pizza, New Haven is home to its own style of it, and it is really, really good. There are a few pretty famous shops, and I’ll leave it to the Yalies to argue over which is the best.
I’m not going to hazard a guess about which of these very fine choices are incrementally better than the others on any one of your criteria.
I’ll just say I have a D at Brown and I attended law school at one of the others, and my D at Brown seriously considered another of your choices, and she was admitted to and we visited yet another.
Personally? I love Brown and College Hill and Providence and strongly prefer Brown’s setup much more than I do the others, with Princeton coming in at a close second.
Yale itself is beautiful, but New Haven is not and my D found it depressing and sketchy. Columbia has a lot to offer, but you have to really want that - urban on steroids - as part of your college experience, and neither my D nor I found that overly appealing. I like Philly overall, and I love the Maine Line area, but West Philly, to me, isn’t that great. I also spend time in Palo Alto on business and find that part of the Bay Area to be antispectic and boring. The campus is lovely … one of the nicest you’ll see anywhere … but PA is nothing to write home about. YMMV.
As a general take, for Poli Sci and preparation for law school you are going to be more than just fine at any of these schools, so I’d focus on the “where will I be happy” stuff.
Lol. Town & Country? Come on. That is the paradigm of suburban sanitized. There are places like that all over now, and likely modeled after that Stepford suburban development. It’s very nice, don’t get me wrong.
To each his own, I guess. Palo Alto is not appealing to me in the least. Not horrible, but not attractive, to me at least.
I like Palo Alto, but IMHO “sanitized” seems like a very apt adjective, at least for the part around campus.
My god, people! Both Stanford and Yale are “best in class” and one isn’t better than the other! Just because you don’t like the urban nature of Yale, or Palo Alto, is entirely irrelevant to the OP!! It’s what the OP likes, and we can’t answer that. The two schools are very different from each other, and the OP just has to decide if they prefer old establishment/centuries of tradition/northeastern/urban or if they prefer new establishment/silicon valley/suburban/California. Over to you, OP!
So, Town & Country is your “beef”? 
When was your last visit to Town & Country? All I know is that I pick up food there maybe once a week and it drives me nuts to get through the parking lot because of the traffic.
Would you prefer more graffiti, trash and homelessness? More grit & grime like say Providence, Princeton, Philly or NY? Missing that slice of NY pizza at 2 AM?
I’d say 99.99% of the people posting haven’t been through the area in years and haven’t seen all the changes, especially since Covid hit.
The secluded comment of Palo Alto is that’s not easily accessible by public transportation, which is true. You could take the train to Redwood City or San Francisco of course, but it’s not as convenient like it would be in NYC say. You’re probably better off taking uber/lyft to those places.
Didn’t say that.
Actually on the strip eating at a restaurant? Probably twice in the last three months. Driving by or through on my way somewhere? Many more times.
These are all preferences, and we like what we like. Providence has a few neighborhoods that would not pass muster to your Palo Alto sensibilities, no doubt. 
But most of it would. I’ll have to turn the question around and ask when was the last time you visited Providence. While not perfect, I find it to be a very nice mid-sized city with a ton of beautiful and old architecture and history. To me, the very definition of character and charm. Much the same to say about Princeton.
Philly and NY … you have to me more specific. Lovely places and areas in each, and some not-so-nice areas too.
We’re getting too far afoot for the OP, but I wouldn’t describe eating at Town & Country as eating “on the strip.” That doesn’t seem to describe Town & Country, as I know it. It’s not a strip.
BTW, I grew up in a large city FWIW.
Hmmm…I’ve never taken the train from Palo Alto to SF. But I just checked, and it takes an hour.
By contrast, it takes around 2 hours from New Haven to NYC.
That being said, I’ve done neither, so, literally and figuratively, YMMV, and perhaps there are others who are more knowledgeable about the actual commute. And that’s assuming OP actually cares about getting to SF or NYC!
I think that’s fair, and borrowing from the federal securities laws, I’ll incorporate by reference that portion of your post into mine.
There are some stunning blocks of homes and other very nice parts of New Haven. But there remains (though greatly improved from years past) blighted parts of it that are, for some, hard to ignore. I’d certainly choose it over Syracuse, NY, but they share some aesthetics in common. Driving to the Yale Bowl will certainly give a prospective student a good view of the other side of New Haven.
But it is important to add, as you did, that New Haven has some beauty to it beyond just the Yale campus. That is true.
I have to been to all these areas except Princeton, and given the OP’s criteria, would pick Columbia (Morningside Heights), followed by Palo Alto, Providence, New Haven, then Philly.
The problem with taking the train to SF is that the last stop is in China Basin about a block from Oracle (SF Giants) Park. So, you’re pretty far away from most of the sites in the City like Union Square, Fisherman’s Wharf, Ferry Building, blah, blah, blah.
You could transfer over to BART at say Millbrae and get yourself closer to “stuff.”
Ultimately, the trains and restaurants are of minimal import to most students. How often did you go out for anything other than pizza or coffee in college? How often did you take a train anywhere during the semester? Maybe once a year during exam reading week, but it’s not your day to day life.
Walk the campuses. Look into the classes and traditions and go where you feel at home. That’s different for everyone, and alums debating whose school is best is pointless.
I would think based on the OP’s age and LGBTQ+ status that expanding the dating pool to NYC and SF isn’t a bad option.
The OP is out of the country, so walking campuses, which I’m not really sure is feasible given on is 3,000 miles away from the others, is not feasible given OP has to reply by May 1’st. This assuming the OP could even afford these trips, which we really shouldn’t assume.
All the time! 
But your point is well taken. MOST of the time, especially in the beginning, is on campus.
I wish I were an alum of any of these schools.
I suppose one thing the OP should weigh is where they see themselves potentially settling down with a job, the Northeast or West Coast? It’s a different culture.
Just to add a bit about public transportation at Stanford, in case that’s of interest to the OP, our son had to use several of them last week when he had to go to plan B for an errand. The Caltrain station is near campus and the Marguerite Shuttle can take you there from campus on weekdays or you can easily bike there. Caltrain runs north/south on the west side of the bay. There is also the Dumbarton Express bus service which goes to the Stanford oval and goes over to the east side of the bay where you can get off at a BART station. He’s learning to use those services to get around without a car and it’s working out well so far. Certainly better than SoCal public transit.
I’ve actually had the privilege of visiting all the campuses. I enjoyed walking around the physical campus of Yale, Brown, and Princeton the most.