(To clarify for the slow witted: I'm neither implying nor assuming parents get random third-party individuals to teach their children, but if you wouldn't trust your child to their educational skills do you really think such people should be allowed to teach their own children? It is still a disservice to society either way.)
Yes, they should be "allowed." Some turn out some very successful people. For others, it doesn't work out for whatever reason. We still certainly need public schools because most parents have to work full time, and there isn't enough time. Homeschooled children still have to pass all the same testing requirements as do public school kids, I'm pretty sure.