While the seller is not breaking the law, the buyer is breaking the law if he is a felon buying a firearm. The crime is still being committed, and the person that should be punished (the one that knows that he is a felon) is still breaking a law that he can be punished for. Our current law still works.
Your proposal would take a person who is doing a completely legal thing - selling a firearm privately, and would turn them into a criminal.
The person selling the firearm has no legal right to look into the private history of the person purchasing the firearm.