I think there would be little problem for the government to ferret out all gun owners. For example, the Attorney General could subpoena the NRA membership lists, under some guise of misuse of funds to influence some politician, All they would need is a complaint. ATF could step up inspections of FFL,s and gather info from 4473. Postal inspectors could compile lists of names of those of us who receive gun magazines. Concealed carry licenses, already part of govt data bases are available. Informants, and the methods are only limited by your imagination.
I remember one case, where we had a child injured in an auto accident. We could not find the parents. I stopped by a car dealership, walked into the finance manager,s office, explained the situation, requested a credit check on the mother, and in three minutes, had all her vitals and the name of her employer. Sure enough she was there at work, and not one neighbor knew her name, or place of employment.