What government do you mean? My understanding is that schools are managed by local boards. You can't get farther from the "government" and closer to people than that.
The local boards often do not have the power to:
(1) Raise revenue that would fund much independently of outside funding,
(2) Freely control how much of funding they do have is spent, because the outside funding sources or superior (e.g., state) law independent of funding sets narrow parameters for many of their decisions.
Districts can raise tens or hundreds of millions of dollars via bonds (requires voter approval but usually passes). They can also pass parcel taxes, which aren't subject to Prop 13 although they can be regressive. Most of their money does still come from the state, but IMO that's not a bad thing, since keeping it more local raises issues with income inequality.
There's very few limits on how they spend their money. I think the only caveat is that districts get extra amounts of ADA money based on the attendance of certain categories of students (e.g. ones that are still learning English). They're supposed to spend that money on relevant programs, and the county has to certify that the spending on those programs lines up with the amount of extra money that is being received.
Couldn't a state, or a district in a state, opt out of Federal funding and then be free from those Federal programs and policies?
Yes, they would lose 10% of funding, but if Federal policies are the reason our schools spend way more per student with worse results than other OECD countries, then getting out from those policies should let them massively improve even with the reduced budget.
[1] https://www.npr.org/2016/04/18/474256366/why-americas-school...
You get functional systems through competition, having many choices so that even if one (or 90%) of them are crap, you can go to the one which isn't. There is by definition no competition at any level of government, which is why so many government-managed services are crap. There is also little competition with most big businesses, which is why big businesses are crap. There is plenty of competition among small businesses, which is why you can go to most walkable downtowns and pick a restaurant and it'll be pretty good, or visit most AirBnB hosts and have a decent stay, or look for a favorite YouTube channel and find one that's interesting.