Aren't school boards elected at a local level in the US?

I don't see how that's supposed to argument for the department of education being well-funded. The department of education does happen to be a federal agency.

but unless Zuckerberg is planning to personally oversee each individual grant given to each individual school or teacher,

I think that sentence illustrates a core bias of the current system. The current system will try to fund schools or teachers while bringing the field forward might also need a lot of investment into elearning.

I think that making public services such as education dependent on the benevolence of rich people with significant political interests

I'm not saying that there should be no government spending in eduction. I'm advocating plurality. Some spending by the government and some by private hands.

Legacy is warm fuzzies. And politicians seek it too, in fact even more than billionaire

Politicians also seek legacy but they are heavily constrained by realpolitik. Mark can give out money to optimize for leaving a legacy in a way that politicians can't.

Masters in education have been shown to be worthless when it comes to teacher performance. Performance metrics on the other hand seem to work.

Currently due to the power of teachers unions people with a masters in eduction get unfairly payed more money. Most schools don't pay well performing teacher more. If you leave it to the department of education that likely won't change. When Mark however gives out grants he's quite free to finance performance-based teacher pay.