Talk of arming teachers is a red herring. Talk of gun control is also a red herring.
The root of the problem is not mentally unstable persons getting access to firearms, the root is a culture that creates so many homicidal mentally unstable people.
I'm not sure you want to go down that path.
#1) I am unaware of any culture that does NOT produce homicidal mentally unstable.
#2) We supposedly live in a free society-- that freedom entails the first amendment which allows freedom of speech.
I think mental health sciences are 100 years away from being able to tell you what it is about "our culture" that produces mentally unstable people. We can easily derail to a 100 pages of nature vs nurture debate on mental health...
But your argument, that is, attributing the root problem to "culture" and advocating we fix our culture is both impossible and would seriously lead to a tyrannical form of government. We'd have to nix the first amendment and I for one do not want to live in a society where freedom of speech/thought is illegal.
So I will argue quite simply that it is beyond our technical and medical capabilities at present to fix the root cause of the problem as you see it, we do not have the ability to fix the culture. So, I would argue that you go the next rung up the root cause ladder, and that would be identifying and preventing mentally unstable people from getting guns. That solution is something that is within our capabilities to do with some reasonable probability of success.
Currently, there is absolutely no linkage whatsoever between mental health facilities, mental health drug prescriptions, requests for gun ownership, and mental health screenings / assessments of people wishing to buy guns. Similarly, there is no ongoing requirement for mental health assessments once you purchase a gun. As a nation, we really don't have a periodic requirement for citizens to get mental health screenings. Would be nice if we were to get a quick and dirty mental health screening every time we needed to renew our driver's license.
Since I'm eating pie-in-the-sky fantasies, I'd also wish for a national health care system enabling periodic mental health screenings when you get your physicals...