My (not data based) impression of school levies is that they nearly always get approved by voters, even in tax-averse areas, so if there is a lack of funding, it is usually real, rather than through a misplaced need to be "efficient".
What gets approved by voters? Ahh, right, government services. How are those paid? By taxes. Who collects taxes? Governments, of course.
I don't know where you are in the world. In the US, public schools are funded by government money counted by number of students and their test scores. So more students = more funding, better scores = more funding. There are other kinds of schools, private schools and charter schools come to mind, with different funding types. But often those include additional costs to the parent on top of the taxes they already pay.
How do public schools get managed by the district? Again I'm not sure where you are, but here the public school administration gets voted in during government elections. The public education system's requirements are defined by law and, above the district level, managed by county or state education services.
> if there is a lack of funding, it is usually real, rather than through a misplaced need to be "efficient"
Don't get me wrong, I think efficiency has its place. But I think it is extremely easy for school administrators to end up in a business-first mindset instead of a serve-people-by-educating-them mindset.
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".
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