There are no restrictions on any other right so long as the free exercise thereof does not harm another person, why should the right to defend ones self have limits? The right to vote is even more important and impactful than the right to self defense, so following your logic, people that are too immature to know when their life is endangered should also be restricted from voting until they can 'prove' their fitness to cast an informed vote. The problem lies in not enforcing the existing laws, not in the lack of laws. It is already illegal for person adjudicated (due process) incompetent to possess a firearm. Remember the Texas church shooter that was stopped by an armed neighbor who chased him down? The STATE failed to report his dishonorable discharge (felony), the STATE failed to report his mental condition to the ATF, and the STATE failed to follow it's own rules and duties to execute the existing enforcement mechanisms that they were obligated to do. Bugs the crap out of me when a person fails to follow procedure and the response from those in charge feel the need to implement even more procedures. It's not the fault of the procedure, it's the fault of the person who didn't follow it.
So, if a deranged dude can't get a semi-auto, he could just get a semi-auto or even a pump shotgun. Pretty sure you can do quite a bit of damage even with a pump shotgun or lever action 30-30. The problem with technology is there is no way to put the genie back in the bottle. 150 years ago, no one could figure out a lightbulb. Now pretty much any competent craftsman could make on if he really needed to. Same with firearms.
Instead of focusing on the symptom, which is what a mass shooting is, we should focus on the cause. The lack of accountability, lack of empathy, and increasing focus on the importance of "me". When the kid finds out the world does not revolve around them, they lack the moral foundation to cope.

If you agree to a background check for voting, or for worshiping, or for speaking, or before you are allowed access to legal representation, or before you are allowed access to public records, or before you are allowed a jury trial, I may entertain your idea for increased restrictions on the 2nd