To clarify...
3 of the major traveling gun shows on the west coast are The Rocky Mountain Gun Show, Collectors West, & Crossroads of the West. All 3 shows allow non FFL's to have tables where they can buy and sell. I've attended the Rocky Mountain shows in Utah, Nevada, & California. The Crossroads shows in Arizona, Utah, & Nevada. And the Collectors West shows in Oregon and Washington.
Every show I have been to in the past 6+ years has had a table specifically for the private collectors and out of State FFL's to run NICS checks. When you register for a table at any of these shows, you sign an agreement that requires you to use these services.
That being said....If "Bob" walks into the show with his rifle over his should and a for sale sign hanging out of the barrel, he can legally sell it to anybody without a NICS check because he is not associated with the show nor is he an FFL holder. THIS is where the "Universal" part comes in and it really has nothing to do with any kind of "gun show loophole." "Bob" could easily place an ad on Backpage and find his buyer there instead. Where you are introduced to the buyer has nothing to do with the "Universal" part but the media wants you to think it's about gun shows.
There is no way to effectively implement a "Universal Background Check" system these days simply because there are so many guns that are already unregistered. The only way to do it would be similar to what they did with the 1934
National Firearms Act. This would require full gun registration for EVERY SINGLE gun in the country and that is never going to happen in the United States. Even today, 79 years after they passed the NFA, there are still "machine guns" that were never added to the registry or destroyed.