I don’t think any reasonable human that owns guns (I am a gun owner) would be against a full background screen on the mental health of potential gun owners.
As for the style of gun - it doesn’t appear as if the old six-shot revolver is the weapon of choice for these cowards.
As long as the AR / machine pistol, etc. is the weapon of choice for the cowards that perpetrate these atrocities, they will be scrutinized.
The perpetrators of these crimes should be kept alive until mental health professionals have exhausted their profile of tests, Profilers have catalogue’d whatever they need to catalog - after that immediate and cost-effective extermination without fanfare.