Well, considering that the biggest period of growth private schools experienced was during desegregation, I would wager that most private schools exist as an escape hatch for parents who find their school district undesirable for one reason or another. Also, most private school students in the US go to a religiously-affiliated institution so there is a strong element of religious indoctrination there.

Stereotypically, charter schools are the ones that are supposed to be better than public schools at a cheaper cost to the government (charter schools can be private or quasi-public).

I'd bet that private schools, on average, have more money per pupil than public ones.