One aspect I have not seen mentioned, and I agree with the culture, good coaching, and talent being the drovers of good programs, but recruiting help for college has not been mentioned.
A private school coach has more time than a public coach, who typically teaches, to spend more time helping his kids get recruited. If a kid has aspirations to play college ball, it's probably a better move to go to a large and successful private school. I think that adds to the talent level being higher than some publics.
About facilities, go south of I80.and you will see some solid programs with really bad facilities, and some terrible programs with great facilities.