Comment by cma256

In many places in the US, though probably not all, school districts levy their own taxes independent of municipal or county governments. The reason we might not describe school boards as "the government" is, well, they don't govern us and are distinct from the organizations which do. They don't have the power to levy taxes or pass laws. Its almost like a large co-op where all the citizens of an area fund and vote on the inner workings of the school.

And, as you say, they are regulated by the government but so is everyone and everything else. In my view, its appropriate to describe public schools as part of "the government" but I also respect the GP's view that they're independent enough to be describe as not "the government".