It won’t accomplish anything, but they’ll all feel better about themselves
However, that prompts this question: just which tragedy would that have averted? In Newtown, the shooter got his weapons the way most criminals do — by stealing them. In fact, a background check thwarted his attempt to buy his own firearms. The Aurora shooter apparently purchased his weapons legally, but didn’t have anything on his record that would have stopped the sale; even assuming that the Gang of Four expand the investigations into mental health for background checks, the only instances of mental-health assessments he had prior to the shooting appear to have been informal. The Tucson shooter had an arrest for drug-paraphernalia possession on his record, but never submitted to a mental-health exam.
Inconvenience millions of law-abiding citizens? Check. Stop shootings? Not so much.