I am not sure "arming teachers," will help if teachers do not want the responsibility but I think if they do want to carry they should be allowed to. Secondly, I do think having an armed guard or guards would certainly help and would absolutely be a selling point for me if I chose to put my child in a private school.
But I think a better question is how these people are accessing school grounds. Perhaps schools need to have entrances and exits that are controlled by gatekeepers. This would be a societal shift as to how students enter and exit school, but that's probably a way easier transition than trying to force through policy that is literally enshrined in a Constitutional amendment.
People can bemoan all they want how "Americans love their guns more than they love these victims," but (a) it's probably partially both true AND justifiable (ie, guns save many lives as well), and (b) until we move off our positions on ideological impasses then both sides are equally to blame. The left loves to spend money. They can't come up with a funding solution to do something about this? They have both houses of congress and the Presidency. Where are their answers besides trying to pass a law that is, rightly or wrongly, obviously on the basis of the recent Heller decision, completely unconstitutional?