It's just that you seem to have no idea about the notion of a public good, and that in some instances the state is there to stop self-interest getting in the way.
I'm sure you don't really mean that all schools should be private and therefore only open to those available to pay? Surely even you have to recognise that an educated population has benefits far and beyond those individuals?
If you mean that we should have complete free choice in schools, then that is kind of what we have at the moment and it is a disastrous idea. Great for the successful schools (according to an imperfect league table) as they will have the sharp elbowed moving postcodes, renting houses and finding religion .... but for the others, it becomes a self-fulfilling prophecy that they will fail.
A couple of people in the catchment go private or fight to get into a neighbouring school, a few other people worry and follow suit, and before you know it's slipped even further down as even more 'conscientious parents' fight to get into a better school. Eventually, there are no bright children with involved parents left, and then it gets closed down or put into special measures. Meanwhile, 2 miles away women are flailing themselves on MN that they live 50 yards from the school with a sibling and still can't get it.
What actually needs to happen, which would have the greatest benefit for the county as a whole, is for there to be no private schools and very little choice. That way, every school would have its fair share of bright, enthusiastic children with affluent, educated parents. The child's ability and the parents' pushiness would make sure the education provided was the best it could be.
But people will always act in self-interest - that is what the state is for (allegedly) and that is what it knows about education.