I like how you assert I didn't go to school at the University level when I hold 2 degrees, one from said University.
Please, tell me more!
Oh, right. So here's the funny thing, if you actually balance for socioeconomic equality you find out that the opposite is true of your assertions. Private schools have smaller samples, but typically higher economic stature which affords more resources to said children earlier, which is an outside influence and has nothing to do with actual school performance. When you weight these things, you realize the education is generally equivalent.
Also, holy shit, of course there's a correlation to going to a private college coming out of a private secondary, that's the point. But it's not the defining factor by any stretch. Also, it's well known that some "highly selective" colleges will admit alumni progeny, and it's of no shock you have higher income families sending their children to these schools (also private schools do have a better network for helping move their students to Universities, but their entire premise rests on trying to do that). But seeing as I've known a fair number of people who went to said colleges, who I should mention came from public school backgrounds, well I'd say you probably want to believe that money and social status is equivalent to education and intelligence.
The absolute biggest issue with public schools is that they have to adhere to ridiculous guidelines that don't fit all people. An example was no child left behind mandating a % increase in test score quality each year, which is obviously impossible. That, and they have to take everyone, which means you have dramatic gaps in quality of students for obvious reasons.