That is already the case with existing background checks. The only thing that keeps it from becoming a registration database is that pro-gun republicans have made it law that the info from these forms which the NICS FBI center receives MUST be destroyed within a certain number of days, and NO data collection can be done from them. That little line item in the law is the only thing stopping them. It's a very delicate and fragile bulwark against centralized government gun registration. The paper forms themselves are required to be stored by the gun dealers, and made available to law enforcement for traces when necessary. So a gun trace now starts with the manufacturer, whose paperwork tells them to which dealer they sent the firearm. Then the dealer's paperwork shows to which individual the firearm was sold. Tens of thousands of these traces are done every year - it's a system that works, prevents abuse, and avoids a centralized gun owner database.