My experience with the NYC school system is that it’s a highly competitive process. Public schools start off zoned, and the best school zones command a premium for rent and home ownership. Private schools are harder in that you have a competitive application process and then a costly tuition. Starting off in a good school bodes well for landing in a top middle school and eventually a top high school.

For public schools:

https://schoolzones.us/nyc/ https://www.schooldigger.com/

NYS tests all their schools. A school which is in the top 10% of the NYS testing would be considered elite. Top 20% is good. The majority of NYC schools are bottom 50%.