My paternal grandfather attended the Jersey City public schools. According to him, only the Jewish kids actually studied. My grandfather's stay at PS whatever was noteworthy only for the fact that he beat up the principal! (Completely justified IMHO, but I will save that story for another time).
The elite schools change their face according to the immigrant population. Now, there are a lot of Asians and Indians, etc. However, another school has popped up out in Brooklyn in the Russian community that has pretty strict standards and there are one or two in Chinatown.
My feeling is not to screw around with this stuff. NYC has been educating kids for a long time. Keep trying to make the system better, but don't take the thing apart.
There is probably no other city in the country that gets more full academic scholarships than NYC.