Do all of that, and anyone who tries to buy a gun without a license can safely be assumed to be up to no good. You don't even need to license firearms themselves, which is unpopular.. just the owner themselves.
Most of these nutters are buying their guns legally. Hell, I bet a lot of them buy their guns not even fully realizing what they will eventually do with them.
Illegal sale/ownership of guns is just an issue for the police to handle. I don't think there is much progress to be made on that front.