I live in NYC but my offspring attended a selective public school, not a private one.
That said, absent special circumstances, I don’t think a second or third tier independent school in the City is going to give her any advantage over a good public in terms of college admissions. As others have said, looking at the college matriculation list isn’t helpful, because you’ve no way of knowing how many celebrities, children of celebrities, developmental cases, legacies, URMs, and recruited athletes are on the lists you see.
The special circumstances might involve ECs, but you say they are equivalent at the 2 schools. If you had a D who excelled at field hockey, fencing, squash, etc., and these aren’t available at your public school, that could make a difference. If you had a pre-Julliard kid, that could make a difference.
Schools also have cultures. It may sound sexist, but especially for a girl, it can be hard to have less money than your classmates. I have a young neighbor, now in her early 30s, who attended one of the very best all-girls schools in NYC. They wore uniforms. But I once jokingly asked her what the “in” brands were, thinking it would make no difference with a uniform. She rapidly recited a list. She named a sneaker brand and then said “but the kids without $ wear Sketchers,” a brand of coat, “but the kids without $ wear ___,” etc. And, believe it or not, she was a nice kid. She was just very, very aware of which girls in her class couldn’t afford the “other” uniform. Most of the girls had weekend homes, and the girls without them sometimes were left out of weekend activities.
Also, a lot of kids in the NYC independent schools use tutors. I’m not talking about kids at the bottom of the class. I’m talking about kids who use tutors to make As in everything. See https://www.nytimes.com/2011/06/08/education/08tutors.html or https://www.buzzfeed.com/buzzfeedshift/career-confidential-the-nyc-tutor-who-helps-rich It can be very hard to keep up if you can’t afford to pay for tutors for your child. (Tutors usually cost $70 or more per hour.) Before I’d enroll my own kid in an independent school, I’d try to find out how common tutoring is at the school, i.e., what percentage of the class is being tutored in one or more subjects.
Suburban schools have cultures too. At some, it’s all about sports and it can be hard for a kid who just isn’t interested in them to fit in. Or at a few, the competition is all about the car you drive. So, again, see what you can find out.
Good luck!