Actually, you do not need to undergo a background check, or do paperwork, to buy a gun from a private individual who is not engaged in the business of regularly dealing in firearms.
So if I want to sell a shotgun that I don't need anymore to a friend or family member or a stranger, I can just do it. So, you CAN but a gun from a dealer at a gun show -- if he's a T-SHIRT dealer, or some such, who does not regularly or frequently but/sell guns. The lie was in the implication that you could do it with a FIREARMS dealer
If you want to restrict the ability of private individuals to sell firearms, then just say so. But, don't use inappropriate language (the lie) to promote that agenda.