There is a pretty easy solution to accidental gun crime. Tie the action to the gun owner. If a kid shoots himself with your gun then the gun owner goes to jail. If someone steals your gun and you don't report it (obviously if your gun is stolen and you report the action then you are no longer culpable) then when they kill someone you pay the same penalty. This one action would cause everyone to rethink their gun ownership. I will probably own a gun at some point over the next decade for hunting, and possibly for personal protection. When I do, I will be absolutely certain to protect my kids from accidental use by having a gun safe and separating the ammo from the gun.
It would only take a few cases of sloppy gun owners going to jail to waken everyone else up. If you own the gun, you are responsible for it. If you can't be responsible for it, then don't own it.