Allowing (not requiring) teachers and admin to qualify to carry at school might have an interesting game theory effect insofar as you’d never know who is armed - like air marshals. The fact that they exist at all means you have to assume every plane has ‘em if you’re planning anything they’d want to stop.
Discussion
Yes, rather than relying on near retirement teachers to actually do the defending, though you’d be surprised what people will do to keep kids safe, the protection would come from the fact that assailants wouldn’t have an easy target. What person but a cowardly psychopath goes into a school and shoots kids? Increase the risk, preclude cowards from participating.
I’m not advocating any carry here, the students could overwhelm the teacher, and some teachers frankly are pretty unstable. Just access and training for those who are psychologically sound and want to participate. Like you said, an assailant would have to roll the dice….maybe it will be the sixty something liberal, but maybe it will be the retired Navy SEAL, who now has 34 rounds of 10mm to fix the problem.