That's true for recent purchases -- but firearms are designed to be robust and withstand hard use. Pretty much everyone (except my dad!) who returned from WW2 came back with a Luger or K-98. My own state has required permits for pistol purchases since the 1960s, but the permits are still paper with multiple carbon copies. The amount of time and effort to track down everything would be astronomical. And guaranteed to be inaccurate -- the state can't even handle annual vehicle registration renewals without an incredibly high error rate.
They will wait for an amnesty period, and then conduct a number of splashy “mini-wacos” against bitter clingers who have expressed the idea that they will resist rather than submit to confiscation.
That will be the tactic of choice, to cow the rest into submission, but instead of submission, it will trigger CW2, if they go that way.