The NICS background check is not a failure per se, the main problem is that not all the computers can speak to each other. If that were solved then you are able to screen people more effectively.
Trump's point seems to be that only the law abiding are the ones being screened and the bad guys are just getting guns the way they've always done it - Steal them or get one from friends and family.
It is just another federal regulation that isn't accomplishing a damn thing.