A large part of the problem is that many of these people exhibit strange behavior or may be mentally ill ( and not clinically diagnosed).... but have not broken the law to have their rights taken from them.
NICS and state background checks should be able to do a better job at ID'ing people prohibited from purchasing... but that takes effort and paperwork, 2 things that states don't care to do in a timely manner.
But i'm not sure y'all realize how little it takes to lose your ability to purchase firearms in some states.