The biggest argument for tax paid, public sector health care and education, is that it should be much cheaper for the customer when the institutions are made efficiently and not for-profit.
A quality education does not have to cost more than $10-20k to produce (per student). So why are current prices in the $100-200k range?
US spending on public education at the K-12 level is close to 15k per year. Your assertion that removing the profit from the system will lead to significantly lower costs is not supported by the data.
Why should education be a market? Why does it make sense for poor people to get worse education? And if you're thinking the government will subsidize it, how is a sufficient subsidy different than public education?
A quality education does not have to cost more than $10-20k to produce (per student). So why are current prices in the $100-200k range?