You already have the corruption in the market. The problem is that with government regulation, you’re giving teeth to the corrupt marketers. They wouldn’t have it otherwise; there would be too much freedom for competition to tell them to go bark at the moon.
As to getting rid of public education, theoretically, you wouldn’t have to do so in a fell swoop. You could phase it out. It would still be painful, but I think it could be done. Although I feel that public education (and public a lot of other things) is more of a symptom than just a problem in and of itself; the problem of people wanting to pass the buck of responsibility to someone else, and someone else being all too eager to take advantage of it.