Teachers are also "local". The resulting LLM will have to be approved nation-wide, which is a whole can of worms. Or do we need multiple LLMs of this kind? How are they going to differ from each other?
Moreover, people will hate this because they'll be aware of it. There will be a government-approved sanitized "LLM for schools" that exhibits particular "correct" and "approved" behavior. Everyone will understand that "pushing back" is one of the purposes of the LLM and that it was made specifically for (indoctrination of) children. What is this, "1984" or whatever other dystopian novel?
Many of the things that may "need" pushback are currently controversial. Can a man be pregnant? "Did the government just explicitly allow my CHILD to talk to this LLM that says such vile things?!" (Whatever the "things" may actually be) I guarantee parents from all political backgrounds are going to be extremely mad.