This may be true for some applicants but IME at least at our BS most kids come from places with very good to excellent local school options, both private and public. In fact, the college lists are often better for the local schools. So it depends very much on where are you coming from, but to assume most BS kids come from places with lousy schools is simply not correct. It may be local, sports, overall boarding school experience or lousy family situation or social drama at your ultra-competitive LPSl, or desire to go to college in the US and therefore do prep school there (for international applicants) that drives kids to go, but it is not the fact that you can’t get great education and get into very good college while living at home.