By now it should be fairly clear that education in practice is also indoctrination. Even if you're not consciously trying, as an educator, you will wind up profoundly influencing the worldview of your students. This is especially true if you're a member of a class that has a decisive advantage in terms of amount of access time, like, say, public school teachers.
But, back to the original question---for what sorts of government is it actually appropriate for the government to have a role in public education?
Certainly, in the case of a monarchy, theocracy, or dictatorship, it is consistent with the form of government for the government to control public education.
But it is quite obviously incorrect for a democracy to control the education of large numbers of its future electorate. Controlling education is to a great extent battlespace control---it defines to a great extent the limits of acceptable discourse. It is obviously inappropriate for a democracy to engage in propaganda about who should win elections. It can be argued pretty strongly that it is inappropriate for a democracy to EVER tell its electorate what to think, since it is supposed to obey them in the classic 'voice of the people is the voice of God' sense. I suppose if you had an actual republic, with limited franchise, it'd be acceptable for the State to educate those without possibility of gaining the franchise, but even there it is problematic. For this reason I advocate the separation of School and State, at least as long as we insist on making the pretense of democracy.
Reading Enemy Propaganda
16 hours ago