Bingo, and the imposition of a registry would inevitably spark our next hot civil war. Best to avoid that....
where the seller would get a signed certificate proving they ran a background check and it's result, but the government wouldn't store anything
Unfortunately, a lot of people are really really bad about keeping records. It would also be open to abuse by the government seizing and disappearing the certificate of someone they wanted to persecute. Heck, the government sucks here, the BATFE's machine gun etc. registry is in horrible shape, and they sometimes abuse that.