Not really. Once a legal firearm moves into the black market (usually through theft) there is no way to track who has it. To think otherwise suggests that you actually expect criminals and other <i>prohibited persons</i> to go through background checks and to register their illegally held weapons.
I didn't say criminals would register their guns. I said a register would show the transition from legal to illegal. It would show who legally owned that weapon last. Some guns make that transition by thief but many are through sales by people knowingly or unknowingly selling to a prohibited person. There is a survey where criminals get there guns that people like to post when the subject of gun shows comes up. Theft is on that list but it is not the majority.
The purpose of background checks is to prevent dealers from selling to prohibited people. Plenty of FFLs have been prosecuted for such sales using their own sales records.The sole purpose of background checks of private transfers (mandated to be made through Federally Licensed Firearms Dealers to insure hard-copy records are kept); and registration is to provide the government with the means to impose ever-stronger rules and regulations over firearms ownership, and in the extreme has led to confiscation.