“Background check - YES”
WHY?
I’m buying a gun. What the hell reason do you have to interfere? Get a warrant or get out of my way!
Background check captures a felon, narrowing his weapon procurement to the cartel underground. It does not capture the law abiding in any manner, or your purchase. It does not register your name, or your newly purchased gun.
Otherwise, felons have the same privileges as you do. Make them have to feel their way through the high risk alleys, in dark, damp places, before they can get a gun from a guy who might break their neck after selling them a gun.
Simple.