People are being nearsighted about this.
“Ban guns”, “arm everyone” - none of this does anything, it’s reactive.
Proactive approach would be to make it hard enough for these people to get guns legally where they’d give up trying. Here’s a hint - mentally stable individuals do not go on shooting sprees, so those that aren’t shouldn’t have access to guns, just like violent ex-cons.
You want a gun? Ok, you’ll need a license (just like drivers) and part of getting one will include a psych evaluation, with, let’s say, a retest every 5 years. Live with other people in the household? They’ll need to get evaluated too.