It's really not that complicated. It is a reverse causation.
Private school kids will tend to do better because...shockingly, their parents will correlate to higher earners, higher IQs and more resources such as tutors, with fewer extreme cases (not many private school kids will have to go home to a crack house while walking home by gangs or meth neighbors in trailer park). Add to that fewer kids per teacher (another benefit of resources)...
There isn't any special sauce outside of that, within the U.S...which is why even U.S. private schools trail many foreign public schools. After that it comes down to culture. America focuses more on stupid shiz like sports and celebrities than glorifying engineers, doctors, accountants, ya know...the people that make the world go round. The same groups many immigrants look up to.
All said, a majority of the most f****ed up kids I ended up knowing went to private school (almost all went to Catholic HS in Knoxville) and I knew a ton of them. Go figure...