Amusingly, we already have background checks
Except that gun show loophole everybody has been trying to close.
The FFL dealers at the gunshows already
run background checks, and have ever since the NCIS system was implemented in 1998, since the law says all FFL dealers must run background checks, regardless
of where the sale is made. That's why Google suggests "myth" following "gun show loophole". I remember when the irritating issue at gun shows was running phone lines for all the vendor booths so they could run the checks.