There's a little bit of actionable legislation there. You didn't fall for my trap where I called out all the shit that is currently on the books with the Brady Bill. Props to you.
The only snags I see are the regs on IDs. Private purchase (the mis-labeled "gun show loophole", gun shows are like boat shows it's dealers selling there not private parties) is extremely tough to regulate. Guns don't have date codes stamped on them so you can't tell who bought what from who where. If I want to sell a rifle to my neighbor on the downlow without a background check, they can't really do shit. The neighbor can just say he bought it before the ban and that's that.
This is currently the norm in places that adopted FFL transfer regulations for all purchases after Newtown. It did nothing except inconvenience legal buyers and enlarge the market for illegal transfers where people with questionable records operate adding to the burden of law enforcement who would rather be apprehending gangbangers rather than guys trading shotguns for skeet shooting.
Also, there is no national registry. Who has what is unknown except in certain jurisdictions where it is required locally and even then they only know what is owned legally. What they have illegally is also unknown. Attempting to keep track of it all by starting one is a huge can of worms as far as anything actionable. Plus, a lot of people just aren't going to. I have guns that shipped with instruction manuals on how to cache them (ah, the 90's AWB panic. how I remember thee). This shit is going to be around forever. Parts and plans abound. People who want them will get them. Here's a guy making an AR out of PBR cans in his garage.
https://youtube.com/watch?v=on1d9Bz34bU
That is the hard way, btw. 80% lowers are $50 bucks and with a set of jigs and a drill are untraceable fully functioning firearms that ship through the mail. Then you have the cap-and-ball stuff that is convertible to cartridge. C&R firearms. And a whole bunch of other shit.
But, yeah, go after those "gun shows" where guys swap stuff in a parking lot. Make them show ID. That'll solve the problem.