> This is likely to be a similar sort of law (or lack of a law saying it isn't allowed), where it may technically be true that this is allowed but realistically you would end up in all sorts of legal trouble for hitting students.

It is not; in fact, the reason states mostly have some explicit treatment in law, is that this waa tested the whole way up to the Supreme Court—Ingraham v. Wright, 430 U.S. 651 (1977)—which found, that, in the absence of specific state law to the contrary, corporal punishment, by public schools, without parental consent or even notice, was Constitutionally permissible. Even people that support corporal punishment in schools often weren’t happy about this. There's no secret additional layer of law beyond the overt ones which will get people in trouble for this.