Generally my take on this is: (1) all schools should select kids at random from applicants and (2) private schools should charge selected kids based on what those families can afford, up to their full tuition which in the Bay Area averages over $30K/year, (3) funding for every school should come from national or at least state level so travesties like Piedmont can't exist explicitly for the purpose of robbing poorer kids, 2 blocks away, from having decent schools (and parks, that's another subject).
Anything else is hyper regressive toward children. F-that!
(sorry to hijack the topic, but this is a subject I deal with a lot and royally pisses me off)