People who -knowingly- sell guns to criminals aren't going to register the guns prior to sale, nor do background checks, and a registry won't do anything to capture guns that are stolen and sold by criminals to criminal associates. There are 300 million plus guns extant, based on the evidence from countries with far more compliant populations and far fewer guns there's no way to register a useful amount of them. The practical cost in time and money isn't worth it.

The burden of proof that a restriction, all criminal laws are restrictions, will work is on the people proposing it. The default setting is liberty, there's no requirement for those opposed to a restriction to prove it won't work. Yet in the case of registries and expanded background checks, all of the data we have, from every country, fails to support the idea that they have any meaningful, measurable effect on crime and violence rates after passage. The people who commit crimes are simply unaffected.