They would have no choice if we registered every gun. The biggest hole in your argument is guns start out as legal until they’re sold then there is no tracking them. After that guns can wind up in anyone’s hands.
Also there is no major gun problems in countries with sensible gun laws, legally purchased or otherwise.
I would like you to explain how registration will solve the gun crimes in the US. Also, which countries have sensible gun laws and what are those laws? How would they apply here?