I think there is an important point here. In the UK we may not have always had strong regulation of guns, we have a long history of registration. This has meant that, when changes have been made to legislation, it is fairly easy for the authorities to know who has what weapons. I believe the same is true in Australia.
Whilst I am no student of the history of gun law in the USA, I suspect you don't have to go very far back to find a situation where there was essentially no regulation of registration requirement at all, and of course it has always been the case that there will be wide variations from state to state.So I am guessing there are probably a great many firearms in the USA which were acquired perfectly lawfully, but are completely unknown to the authorities. Nothing to criticise there, just a reflection of the very different history of our countries. Perhaps Artie, or one of our other contributors across the pond could comment.
The issue being that if that is the case then even if the US authorities wanted to introduce the sort of measures we have, it would be virtually impossible to do so from a practical point of view.