This is a solved problem, other countries do good background checks by requiring a reason for gun ownership. Collecting (maybe you can't buy as much ammo), range club (they need to vouch for you and you buy ammo at the range), hunting (you have a subset of hunting rifles allowed), self-defense (again, how much ammo do you need for self defense).

I'd like to see the Feds designate all schools crime as federal crimes since they interfere with the civil rights of students. Then make school shootings automatic death penalty, unauthorized guns would have a hefty sentence like they did with drug dealers.