The public school board is an elected body; you can show up at your city or town or village's election and pick the members you like. In practice, few voters are invested parents, and nobody pays attention to school-board candidates most of the time. (If you follow the news, Loudoun County, VA is a notable recent example of how things can go wrong.) So many public school districts are stuck with the same corruption that occurs in any public position.

The Catholic schools' policy on things like this, at least near me, is dictated mostly by the diocese. The state can withhold funding for things like special-ed programs or meals if they won't play ball, but they get more autonomy as long as they aren't violating civil-rights law.