I agree with all that you said, although I do think that teachers should be armed … not all of them of course not. I would guess that most schools have at least 3 or 4 teachers who may be familiar with firearms or even have a CC license already. You don’t need every teacher to be armed to have a good deterrent.
Another plus side I think would be that the teachers have an interest in the students who they have taught for years. They know them and they are not going to let some bastard psycho shoot them all while they linger outside in safety.