> Ideally their kid should still have access to public school education, with all the tools needed to go on to become successful themselves.

Ideally, but in reality public education is not an equalizer like that. Wealthy parents give their students advantages that school can't provide in our society: e.g. upper class culture and habits, resources for extra education, connections, etc.

Even the institutional bits aren't equal: local property tax based funding mechanisms mean richer people live in better-resourced school districts than poorer people, with better educational opportunities.