Well, regardless of possible benefits, I'm pretty against the idea of privatizing schools. Making money will be the main goal, which shouldn't be.
My wife worked at a private school in Nova Scotia that, on the surface, anyway, was supposed to be for kids with autism and special needs. I won't get into the details but it was a total fleece job on the parents, but maintains a legitimate and even somewhat upscale image.
I'm sure there are plenty of good private schools out there, but of course, it'll be the wealthy that benefit... No, I don't see any good coming from an all-private school system. It's not the answer.