Max, good questions. On the first, it is because the poor could get stuck with low-quality schools that these markets should be regulated. There is asymmetric information in these markets (parents don't know the quality of the school as well as the school does), and there are ways of regulating such markets (transparency of information about quality, etc) to avoid the low-level equilibrium.
On the second, the rationale for subsidizing education is the externality associated with education. That externality exists whether or not you regulated the market. The rationale for regulating the market is that, even with the subsidy, the education market may be distorted (for reasons of asymmetric information and others).
Your last point is, of course, the crucial one. I'm hoping that debates like this can at least help us make progress in dislodging the low-level equilibrium and get poor people the quality education they are due.