Yes --- there would still be a need for schools. Cooperatives, private schools and if the community decides --- public schools. Public schools were local government controlled and funded. Many neighborhoods would want a good public school because it tends to increase home values. You could purchase your home in an area with no schools and save the tax money but you could decide to live where the schools are good but you'd have to pay.
I'm on the same page with you here. Keep in mind that *states* generally have compulsory attendance (and the compulsory adoption of school districts) in their state constitutions.
I see that as being within the rights of states, and an expression of state sovereignity. If people don't want to live in a high-tax state, they're free to move to a low-tax state.
I'm also with you that private schools in general should be unregulated, but we do have to deal with the question of militant Islamic schools teaching jihad. That wasn't an issue 20 years ago when many of these private school / homeschooling issues were being hammered out.