Edit: You are also being selective in your examples. Why not compare Berkley to the University of Phoenix? You have to consider the entire range of private schools, and the overall output. But I'm not as concerned with higher education, I think that is a different system with a different goal.
Primary schools need to educate everyone, as a minimal level of what our society needs to move in a positive direction. Private schools would be incentivized to abandon problem kids that hurt their bottom line by screwing up their average scores and what not. This is even happening in public schools with some of the stupid testing schemes we've implemented, but it would be far worse imo with private schools.
Sadly, many kids in special ed programs and many behavior problem kids would not have problems in a different type of learning scenario... but the entrenched K-12 classroom, 30 students one teacher, hour long periods, boring books, exercises, silence, etc., is just a horrible fit for them.
http://en.wikipedia.org/wiki/Phillips_Academy http://en.wikipedia.org/wiki/Phillips_Exeter_Academy
[1]http://en.wikipedia.org/wiki/Ivy_league