If you are moving to NYC with school aged children you might as well know another ugly truth about the NYC metro area regarding education.
Generally taxes are lower in NYC but the public schools
in many areas leave much to be desired, in fact many would call the locally zoned grade, middle and high schools horrible. Parents with means send their children to private schools the cost of which offsets what they are paying in lower taxes.
OTHO one can move to Westchester, New Jersey, Long Island and pay more in taxes but live in a area where the locally zoned public schools are good to great to excellent.
In Manhattan the handful of great to excellent public grade to middle schools are always full with waiting lists. However that does not stop middle or working class parents from "working" the system to get their children in, sometimes it works but often it does not. Of the NYC high schools the top tier such as Stuyvesant enterance depends upon exam scores.
So what this all boils down to is yes, you might find a great apartment/house in say Williamsburg or Red Hook but you probably won't be sending your children to the locally zoned school. Indeed the "problem" with the gentrification by middle class and or upscale families (mainly white) to parts of Brooklyn and Queens is that by and large they still return to Manhattan for their healthcare needs and often do not send their children to local public schools if they can afford different.
One of the first questions anyone with children will ask a real estate agent here in NYC (or find out one their own) is the quality of the local public schools. Parents often base their real estate choices based upon the education system. Private school tuition and expenses like every thing else here cost dear, and not every family can afford that sort of outlay especially if they have more than one child so they hunt out the best public school options.
In parts of Manhattan at least (Upper East and Westsides, The Village, etc..) so many parents sent their children to private schools that the local public ones (which are often excellent) have slots open. Well they did anyway until Manhattan became the new suburb and word got out. Now out of area parents try to seek waivers to get their children in and or use a variety of other tricks.