The form 4473 form gets stored away in the dealer's filing cabinet. End of line.
The background check data gets purged frequently, and never has any information about the gun you're buying anyway. No serial number or anything like that.
So, in most states anyway, there is really nothing to "tie you to" a certain gun or to identify you as a firearm owner, except for a piece of paper in a filing cabinet in the back room of your dealer's store.
...
The firearms registration trace that so many people have been programmed by television to believe is a matter of one phone call really works like this:
1) IF the police come into possession of a gun, let's say a gun dropped at a crime scene, they know what kind of gun it is and have a serial number.
2) They take it back to HQ and can run it through the FBI's NCIC database to see if someone had reported that gun as being stolen. (Guns are only in the database if at some point a legitimate owner has reported a gun theft and their local police put it into the database.)
3) If no hits with NCIC, they call up the manufacturer and ask them to look up the serial number of that gun. The manufacturer calls them back when they've got the info and tells them, "That gun was sold in 1987 to G&A Gun Distributors, in Boise, ID."
4) They then call G&A Gun Distributors and asks them to find the records on that gun.
5) G&A calls them back later and says, "That gun was sold in 1988 to Bob & Jane's Gun Shop in Cedar Rapids, MI."
6) Then they call old Bob and Jane. Now, 1988 was more than 20 years ago, so Bob and Jane can legitimately say, "We burn all our records after 20 years and don't have that info." End of line. --OR-- Bob and Jane say, "Yes, we found the 4473 on that gun. It was sold to Jim Smith, who lived here in town, on August 27th of that year."
7) THEN, the investigators can try to call Jim Smith. Now Jim may have died. He may have moved out of the state. He may have sold the gun to a private party who's name he doesn't recall. All of those are more or less complete dead ends.
8) Maybe though, Jim sold that gun to a guy named Ed who still lives at 123 Street Rd., Townsville, MI. Or maybe he sold it to a gun shop somewhere. So the trail may still, perhaps, be warm.
But each step make the likelihood of the gun being traced to subsequent steps less and less probable.