Based on your spelling, I assumed you were Canadian.
The US has a federal system, not a unitary system. Each state and territory is its own semi autonomous sovereign (to some degree, not to the degree Bundy seems to think). For example, I worked with a guy who was wanted in PA for grand theft auto. Since PA was not willing to pay to extradite him back, they didn't ask Wyoming to pick him up so they didn't have a duty to care. He eventually went back and turned himself in. Of course, the feds can enforce federal law anywhere. For example, local police respond to bank and gun store robberies as robberies. The FBI and ATF will also investigate because robbing a bank or a gun store are also federal crimes. Possessing a legal machine gun in the wilderness is not a federal crime, so the BLM, if on federal land, would not have any jurisdiction. However, should they come across a Wyoming game warden or county police and mention "I saw this guy" your day started to suck because it is a crime in Wyoming.
From what I've read, about 40% of gun sales go through gun shows. That means there is no background check.
The quote was that 40 percent are private sales without BGCs, the claim was based on a study in 1993, before the law was passed. If the seller at a gun show is a federally licensed dealer, there must be a background check. If between two private residents of the same state, then it is up to the state law or the gun show promoter's policy. Some states do require one, or have a licensing mechanism, and some promoters will have a designated license holder to do BGC for private sales.
Whether it would violate teh Commerce Clause, I don't know. As I'm British, the minutae of American law is unknown to me.
The Constitution's Commerce Clause and the 10 Amendment limits what the federal government can do, and what is defined by interstate commerce. If I sell a gun to another Wyoming resident without going through a licensed dealer is legal. If I sell one to Ted Nugent without going through a Texas licensed dealer or Sarah Palin without going through one in Alaska, I would have committed a federal crime.
My intention with teh safety test is simply to ensure that everyone buying a gun has a sense of proper safety practices. Ironically, while the NRA would be dead-set against it, such a test would copy a lot of the content from their safety classes which are rather good. Something akin to the Canadian minors permit test would be ideal. I'm not interested in proposing a test that everyone would fail, just with ensuring decent safety practice.
Yet the manufactures lobby, the NSSF, would probably be for it. The NRA does a lot of stupid stuff that I don't see the logic behind. They alienate most gun owners with the speakers, and the choices of music at the conventions. It is like they even believe the "old white rural guy" meme. Right now the NRA is pushing a bill that would force states to accept carry permits from any other states. Assuming manages to ever pass, which it won't, it would be, and should be, struck down as violating the 10th Amendment.
I'm aware that there are very few privately held full-autos but since, as I said, we're throwing everything out and starting again, we would need to put that ban back into place.
the reason for the large tax and extensive background check in the 1930s instead of a ban is because the backers of the law thought a ban would be ruled unconstitutional. Since the UK has more machine gun crimes than we do (and the rare one isn't done with legal antiques) I don't see the value of the law. While I'm not an anarchist, I believe restrictions should exist if, and only if, there is a demonstrable compelling public interest, meaning there must be empirical evidence that the restrictions on individuals are justified. I don't see it there. I'm not saying we should repeal NFA, but I don't see the social value of the ban.