Background checks are already universal as long as the seller is
a firearms dealer. Regardless of where the sale is made, at the
store, at a gun show, wherever. If you're a dealer selling to
a non-dealer, then a background check is done each and every time.
Hell, even if you're selling to another dealer, you have
paperwork to process.

The only time background checks are NOT done is private sales.
Eg: Private party to private party transactions.

Guess what ? Forcing all sales to go through a licensed FFL
won't do much either. Recall the event that put all this
silliness in motion: The weapons used were STOLEN from a
person who purchased them legally to begin with.

Create all the rules you like, but the criminal mind WILL find
ways around them.

There are only two ways to prevent gun violence against people.

1) Remove all guns from the world
2) Remove all people from the world

Both are about as likely as the other.