Do you need every teacher, student, and support staff to decide on every budget item? No. There are ways of organizing large groups without electing boards and representatives. I won’t deny schooling would need to be restructured, but I think school needs restructuring anyways.
Groups within the school (however they want to organize themselves) could elect delegates (basically messengers) to reach decisions. That is how Europe developed its continental railway system (Kropotkin talks about it in Conquest of Bread). Its also a situation of let them figure out how they want to organize. I don’t think people not involved in the school should be deciding how a school is run. People should be free to organize themselves through voluntary association, not have decisions forced on them. Its about consent, and I can tell you right now, teachers typically do not like school boards.