I don’t believe too many criminals subject themselves to a background check, so doing them will not work.
Judges always want to give thugs a slap on the wrist because most of them are young, but the problem is that they're already hardened killers and letting them loose on the population after a few months or even a couple of years in jail does nothing to protect us.
Whenever I see one of those articles about some thug who has "accidentally" killed a child in a drive by shooting of one of his gang rivals, I always see that said thug had already served not one but numerous sentences for earlier offenses but was back on the street to kill again.
I agree that blackmarket weapons should be hard to get and that stopping weapons smuggling should be a priority, but the problem is really the people who use them, and we're not dealing with them very effectively at all.