It seems to me that you have answered my concerns. All of the background checks, gun control, mental health checks, etc are all good, but many times they fail to ferret out the danger cases. And you can't blame the teachers. Go through a day with a teacher sometime and try to cover all the material on the lesson plans to five classes, take a 15 min. lunch break (if you find the time), and at 3:30 p.m. (if you don't have bus duty), and see if you are tired and drained.
Take a hypothetical case, and be honest. Suppose you are a loving dad with 2 daughters at a school in your home town. You pay, you work hard, your lovely wife is home more than you, and you are well known and respected in the community, participate in the community affairs, go to PTA meetings, and all the rest.
On a certain day, while you are feeling comfortable and confidant that your girls are safe and being cared (remember that the teachers abide by the (in loco parentis principle)). But on the radio you hear a disconcerting report that there might be an active shooter at the school. Your school has adopted the rule of armed staff. Now suppose I am one of those staff members, and that you know my background - I'm an army veteran, I've had sniper training, an expert rifleman, and you and I have discussed the possibility that there might be an incident someday. I have given you my assurance that I will do everything possible to protect the students - even giving my life is necessary.
You go into a panic, and obviously scurry to the school, but on the radio the news comes that the shooter has been wounded by an armed teacher and is in custody. How do you feel? (and be honest)
Your answer will depend upon whether you agree or disagree with me. (some of the above was once correct for me)