Well, sort of. I don't know the specific content of these posts but telling voters they should vote one way or another by a government agency is likely to run afoul of election laws, at least it would in my state, but I don't know KY.

A school is not allowed to tell voters to support a school budget?

You're doing it wrong. They are a government agency. They must allow it unless they can show that they don't have to. That is the precedent we need to find here.

Says who? You can't make them do a darn thing without a court order. (like not delete comments).

You won't find one that says they can. Or maybe you will but until you do, they can't.

Right. And in the absence of a court that says otherwise they can do what they want. There is no other mechanism of enforcement. If they defied a court order that's one thing but that would require a court to rule on the subject.

Your points 1 - 3 aren't without any merit. But until a Court says otherwise it's not germane.