Even if you dislike markets as a mechanism, education is very tricky because it's very very expensive, and prone to important moral hazards when it's free.
Education is not inherently expensive. The student loan mechanisms are what allowed the vast expansion of colleges as capital intensive enterprises. You can find a multitude of discussion about how cheap both public and private education was in, say, the 1960s. And that was with primarily state/institutional regulation. Which is to say the student loans were moral hazard crack to compared to state regulation.
For most subjects, one needs an informed teacher, a class room and some books. Somehow this costs vast sums now. Guess why?