Public schools are a really nice idea. The government builds a building, right in your neighborhood, where anyone can send their kids to get an education for free. It’s simple and appealing.
But in practice, it’s quite a bit different. Land that is in the intake zone for a good school becomes more expensive, and you create a situation in which the school is open exclusively to the “public” of people who can afford a very expensive house.
Look at this chart showing the correlation between the price of a family-size house and the reading proficiency scores in the local school (the outlier, Garrison, where the reading scores are terrible and the houses are expensive anyway is my neighborhood public school):