The only concern I'd have would be with religious schools which might teach BS instead of real stuff. But an appropriate level of government intervention in setting the curriculum and standard examinations would hopefully negate this problem.
"Final question for one million points: roughly how old is the Earth, anyway?"
Let me explain how schools work where I come from. The final year of high school is spent largely doing a series of examinations standardized across the state, called the Higher School Certificate. (It's a bit more complicated than that, there's a school-based contribution to the exams which gets normalized with another set of exams, but let's just pretend it's one big set of exams now, which tests everything you learn in your last two years of school.) The marks are all tallied up, and every student in the state recieves a grade between 0 and 100 which ranks exactly how well they did in their exams (if you get 95.4 then you did better than 95.4% of the state, et cetera). These numbers (called Tertiary Entrance Rankings when I went through, though I think they've changed name several times since then) determine what courses and universities you can get into (none of this namby-pamby admissions essay slash extracurricular bullshit stuff like in the US)... or for those who don't go to university it will be a major line item on your CV when you start looking for work, so obviously all students are very keen on making sure they get a high TER. This provides an obvious point for competition between schools -- schools will compete between themselves to get the largest number of high TERs, and parents choosing schools for their children will look at this information when choosing schools.
It's heartless, and it's cut-throat, and it's incredibly stressful for the students, and it works great.
I seriously doubt it. Just look at all the sham private colleges that exist right now. Also, I think the problem with our kids isn't the school system, but a culture that doesn't value hard work and education.
http://en.wikipedia.org/wiki/Phillips_Academy http://en.wikipedia.org/wiki/Phillips_Exeter_Academy
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.
I think your worry about someone starting a school that is actually a sham are very much out of proportion. In today's public schools, many kids can't read or do basic math, so it's hard to imagine something worse.
Also, the monopoly system is the only reason parents can't easily switch their kid to a different school. If this were an option, parents would be much more critical. Most parents today just settle for whatever is publicly available b/c it's "free" to them and they'd have to move to a different district to put their kid in a different school.