Yes, it is quite as you say, sometimes there are other non-economic incentives that encourage public property and not private, without taking into account the profitability, this would be the case of the police or military forces, or the system of justice, to name a few. However, I believe that, if the State dominates education, regardless of whether it is profitable or not, it is more likely that by simple inertia it will end up becoming indoctrination. I am one of those who believes that the ability to reason is not granted by any university or school in particular, I feel that these institutions should only exist to teach technical work and specialized knowledge that allows them to integrate effectively into the work environment, ie , an education focused on labor integration and the production of wealth. For the rest, I believe that each group of people should have the power to create their own educational institution, and give themselves their own curriculum, however, we must be aware of the danger of this, since as all freedom, represents a responsibility and a risk, because there are very powerful people, with big interests and a lot of money, then, they could take advantage of the vacuum left by the State to indirectly monopolize education, which would be a big problem. While there are small powerful groups against social health, it will be very difficult to make anything that tries to solve a social problem work properly.
RE: The reason why public education does not work