> those who break the law with the guns they currently have
Those who break the law regularly have the means & connections to buy a gun illegally.
I very much doubt distressed teenagers will have the money & connections to buy a gun illegally especially on short notice.
Definitely won't stop all crime (just like it doesn't even in countries where guns are illegal) nor resourceful, determined attackers, but would definitely cut down on crazy teenagers doing mass shootings.