One of the proposals I've put forward in lieu of background checks would be this:

1.) If you're a prohibited person, your driver's license/state ID is conspicuously marked in such a way as to show that you're not allowed to have a gun.

2.) Make it a legal requirement that in all F2F transactions, the seller must check the buyer's ID for that mark. No mark? Good to go. Didn't check, or sold a gun to a person with a marked ID, and you get caught? Go to jail.

Such a system would be much faster, cheaper, and more equitable than forcing everyone to go through a centralized background check process. Additionally, it would work to empower gun owners to act as the first line of defense in keeping guns out of the hands of those who shouldn't have them.

But, since BGCs aren't about preventing crime, you'd never see such a system get any sort of traction.