Here's the bottomline on NYC education. It's a meritocracy.
If you are an extremely bright kid, then you get a free ride to some of the best private schools in the world that lasts through college. Nobody much likes to talk about this, but there is a mechanism in place to track those kids out of public schools and into the high end private schools for free.
If you are average bright, then you get to go to schools like Bronx Science or Stuy. These are the high schools that have ten or fifteen nobel prize winners in science along with graduates like Lucy Lieu, Tim Robbins, Thelonius Monk, Isaac Asimov etc.
If you're a kid that is just average, your parents would do well to move to the suburbs.
I have a friend, or rather an acquaintance, who is one of the people in New York who puts up big money to educate talented kids. So I'm somewhat familiar with what you say. And I'm familiar with the elite public schools. But you can be very, very bright and still not get in if you aren't also lucky.