I suspect the phenomenon of paying for an education one doesn't really find satisfactory, but ostensibly chose (and then paying extra to get the education one wants) plays out to some extent at least a bit further down the income scale too--and likely for related reasons. aquinas's observation rings true--they are buying not an education, but access to something else (Ivy enrollment, social status, a particular community), and the education per se is sacrificed to that other goal. Even high-performing suburban public school districts are subject to the same pattern, but instead of paying a specified tuition for an undesirable educational product, families pay higher property taxes in exchange for access to a certain kind of "free" public education--often also with markers that provide increased access to elite post-secondary institutions, social status and communities with certain qualities. And then on top of property taxes, parents pay for afterschooling and tutoring, to get the education they think their children need.
This isn't to disparage private schools, high-performing public schools, or the families who choose them, but simply to note that, for most affluent families, there is choice involved. And those choices could be wielded in different ways, whether to prioritize education or other values, and using different approaches to achieve their goals. Paying for elite private or elite public schools is not the only way to invest financial resources in high quality education.
And btw, private teacher salaries, on the average, are lower than public school teacher salaries. And typically have smaller benefit packages. If one funded a private school at public school teacher salaries with benefits, one could likely attract some quality staff.
What public school districts spend per student:
https://www.census.gov/newsroom/press-releases/2019/school-spending.html
