Why would the government need to know who owns what? How about a background check that merely tells the seller whether you are OK to buy the weapon. The government has no idea what you bought.
That sort of law I would be behind 100%. It checks the buyers and doesn't build up a data base of who owns what. Lists get used. That is a fundamental law of government and after the revelations of all the illegal wiretapping and other criminal behavior they have been caught doing.
If a gun is used in the commission of a crime, one critical piece of information is the ownership of the gun. Quite often the gun owner, family member or friend is the perpetrator. Although ATF records are far from complete and many records lack ownership information, they do over 335,000 gun tracings a year for law enforce. So yes, it is of value. If it wasn't the police wouldn't be requesting the information.