My opinions may be colored by the fact that I am a single man who is PO'd that I have to send tax dollars to schools nobody in their right mind would allow their children to attend. I can't think of one area I have lived in (New York, Illinois, Florida, Washington, NJ) where the public schools were mediocre at best, even in wealthy areas.
Scarsdale apparently has pretty good public schools.
Also, some of the elite public schools in NYC rival or surpass most private schools.
But let's be frank here. Public schools in NYC have always served two purposes. They educate those who are highly motivated (but without money) to take leadership positions in their fields and they serve as the first step of integrating new immigrants into American society.
Oh yeah, and don't get me started on how many Nobel Prize winners there are just between Bronx Science and Stuyvesant.