In my area, amongst my acquaintance, families who can afford it tend to live in towns with excellent public schools and keep their children in public school. If they have a less successful child who they fear is being overlooked or simply not getting the attention or structure they need, they may send them to private/parochial. The kids in need of attention are generally sent to private, the kids in need of discipline/structure are generally sent to Catholic. There are other people who went to private school as children, who have family money, and who simply assume that kids go to private school if their families can afford it and never even try public schools. Those families may choose to live in the same comparatively upscale towns or in more urban areas where the public schools aren’t great. It doesn’t matter to them. There are Catholic families who simply assume that the kids will go to Catholic school if at all possible. There are Jewish families that assume the same regarding Jewish schools. There are Christian families who either home school or send their children to Christian private schools. There are some people who have money and are into the Waldorf school way of thinking.
In my experience, most people in the US with kids try, if they can, to live in the best public school district they can. Usually that means they spend more money on housing. If the OP’s family is spending as much as they can to live in Scarsdale or a similar school district, they are already investing heavily in their kids’ education. IMHO, if you are planning to send your kids to schools like HM–assuming that they can get in, which is a big assumption, no matter how bright they are–there is no reason to spend extra $$ to live in an expensive school district. Of course, there is always the matter of keeping your options open.
When we moved here, we deliberately moved to a town where we felt we could send our kid to public school, even though at that time he was in private.