If government is not involved in education "at all", then you have the situation that people pay $$ for schools and then have no say in how the money is spent. This would not go down well...
Education is the vegetables on the dinner plate of life. I is a complicated long term investment to make good choices, but those choices have huge externalities. Parents make choices that affect their children (that is an externality) and those children grow up to affect society (another externality)