An interesting thought experiment. The most obvious flaw in it is that while people have to eat, they don't necessarily have to be educated. Not every child is born into a family with parents who, in a free market, would devote lots of resources (time, money) to ensure their children are educated.
The whole voucher for private school concept does not seem unreasonable to me as an alternative. However, the notion that we can just stop taxing people to fund education on some level is ludicrous. A free market in education will only work in families with parents who care.
I'm not willing to withhold opportunities to children who made the mistake of not being born into a family that values education. And if that means the rest of society has to pay to educate those children, I'm OK with that. An educated society is to the benefit of everyone in that society.