Add to that that "private schools don't have the discipline issues publics do" If you as a parent have to pay $20,000 or more for a year of high school tuition, you are probably a fairly involved parent and on top of things. Obv not everybody pays full tuition.Also explains the mythical 'superiority' of private schools.
In scholastics and athletics if you can pick your team you're going to do well regardless.