in the hands of untrained people with no background checks

In the context and language of the time, "well regulated" essentially meant "properly trained in their use" so there's a constitutional argument to be made for mandatory firearms training as a prerequisite gun ownership. That's...not going to reduce shootings in the ways you might expect, since a large portion of shootings are not done with legally owned guns to begin with, and there's no good reason organized crime (think gang and/or narcotics related, connected is why a few dozen counties account for something like 4 in 10 homicides) would use guns legally registered to themselves.

Would probably reduce the suicide rate though - anything that breaks the immediacy between wanting to do it and trying to do it tends to.

An important step up on the whole issue would be making gun owners legally liable for any crimes committed with a firearm they own, even if it's stolen unless the theft was reported beforehand and there is clear evidence that it was properly secured at the time of theft. You steal your father's rifle and go shoot up your school and he's on the hook for the murders too. Would make a bunch of folks much more concerned about properly securing their arms, which would further reduce the suicide rate if nothing else and put a dent in the "lone nutter shoots up their school" types as well.