I'm looking for ideas as well, and so far the (concealed) arming of teachers/administrator (that volunteer, of course) is the most effective idea I've heard so far. It has the advantage of being relatively low cost (training and perhaps covering equipment costs, versus hiring full-time guards or reimbursing the police department), the teachers/admins know their school layout and people that should or shouldn't be there better than anyone, and because they are carrying concealed (and there's presumably more than one of them), there's no one wearing a uniform that says "shoot me first" to any prospective murderer. Tactically, some teachers might hole up with their students in locked-down classrooms, others (or administrators) might go towards the sounds of gunfire. But more significantly, they'd turn schools from soft targets to hard ones in the mind of a prospective murderer - certainly no "outsiders" would have to know if there were no teachers/administrators in any particular school actually armed under the program or not. The question of mentally ill teachers is a relatively easy one - we already try hard (I assume) to make sure our children are not being taught by the mentally ill, but in any case, what is the difference between someone mentally ill carrying a firearm, or someone mentally ill needing to go out to their car to get a firearm?
Again, I don't oppose or support the idea. But almost everything you said here dismisses as insignificant some rather enormous issues.
Any training of teachers/administrators would be an additional cost for schools. Who pays for that? In this country schools are under a full frontal attack from legislators to cut their funding. Maintenance training for firearms handling for officials that sometime carry incidental to their job (not LEOs but bailiffs, game wardens, and what not) is somewhere around 80 hours a year. That does not include initial training and retraining on tactics etc. (To be effective a teacher would have to know what tactics the police will be using so that they don't become a problem during a response.) That 80 hours is an enormous expense that someone has to pay for.
Teachers don't necessarily know who should and shouldn't be there. But even if they did, many of the recent situations have involved people who "belonged" where they committed their crimes.
"There's no one wearing a uniform that says shoot me first?" Um, being the only adult in the room combined with the knowledge that any teacher may be armed sends that very message!
As for holing up in a classroom behind a locked door. You should inform yourself about this. I am personally aware of a few schools where the ability of teachers to lock the doors has been taken away recently because a) it cost money to deal with lockouts, and b) the police recommend it so that they don't have to deal with locked doors during any response. I find the second reason particularly stupid. But the point remains, something as simple as that can be quite expensive!
The "program" you suggest sounds similar to what we have for air security and teachers would be like air marshals. It costs billions of dollars and relies on extensive and invasive individual screening of every person. And few people are convinced it is actually effective.
Again, I don't disagree with you and I don't even think you are wrong. I just think that what you suggest is enormously expensive and full of complications that make it unworkable.