Looking at the uproar about Critical Race Theory, it appears that even the deepest red states have school hierarchies run by the blue. In addition, the Department of Education and other federal agencies are a thing.
This is definitely not the case. Consider: unless consistently-"red" legislatures and governor's offices are putting "blue" people in charge of their educational departments, it couldn't be the case. Further, in many states, individual districts play a major role in deciding the course of education, so even if that were the case, "red" school districts would have to be electing "blue" school boards, and I can 100% confirm that, as one might expect, they do not do this, generally.
[EDIT] further—and I'm heading into speculative territory here, I admit—I'd guess that people holding school administration PhDs—so, qualified for superintendent-type roles—identify, on average, as farther right than the average of PhD holders, and probably than most individual PhD areas of study except maybe business-related ones. There are lots of very conservative school administrators out there.
Secondly, these sorts of policies often come from the bottom up, from students or teachers. Administrations often roll over out of self preservation or apathy.
> Secondly, these sorts of policies often come from the bottom up, from students or teachers. Administrations often roll over out of self preservation or apathy.
Is happening in any widespread or common way in k-12 school districts, with admin hired by "red" school boards elected by "red" voters capitulating to most any "blue" political inclinations of teachers. As for the students in those age ranges, for most of that span they have a strong tendency to very closely follow the political views of their parents, so that's mostly irrelevant as far as pressure goes.