I think that it is obvious that UBC can lead to tracking individuals depending on how it is structured. Gun controllers and to some extent law enforcement want this information and obviously those who own guns mostly oppose giving them this information. Thus, we have been at an impasse on the issue. Given today's technology, automated systems could allow individuals to determine whether someone is eligible to buy a firearm without recording the specifics of the transaction. Think of something like Paypal where the buyer and seller both get confirmation of the transfer of money (in a loose sense) without the specifics of a particular credit card etc. But the system would have to employ some way of forgetting the record after the inquiry. However, the pro-gun forces in Congress lack the power to force the issue and those seeking more government information on who owns guns are unlikely to compromise their demands for everyone's identity. Moderates in the issue go back and forth depending on the news of the day.
In the absence of any great societal consensus, government policy is being set at the margins with almost random efforts to show "something" is being done. If you think of it as ritualistic, it would be correct--e.g. Connecticut and New York laws have very large non-compliance ratios. If a new event happens, then further "restrictions" will be passed with little effort to ensure compliance. Forcing compliance will cause a major backlash in politics and so is avoided for now. The political ritual in these states apparently demand "something" must be done and the governments in those states comply. Every nation-state or empire goes through similar problems, some eras stress centralization while others decentralization. When fundamental agreement of what a state's government should and should not do become splintered among the population, this type of disorder (think legal anarchy) spreads. We see this on immigration, abortion, the role of religion in society, use of drugs, immigration, the role of the military, law enforcement, government spending and taxes, etc. and should not be surprised when it involves the topic of firearms as well. Washington is deadlocked on a lot of issues right now because voters and elites are split in attitudes. Thus, the conflict spreads to the states with more homogeneous state governments taking positions almost diametrically opposed and those in the middle fluctuating.
Repeatedly, we have seen that government can abuse information gathering systems such as using them illegally to track love interests, political enemies, etc. Some things like the printed 4473 obviously hinder this process by making it time consuming and more difficult. This serves as an incentive to only do so when needed for LEO. Make it easy, they will follow the path of least resistance such as the automated license plate readers and using traffic cams to track individuals of interest to law enforcement. Currently, the Supreme Court has a very important case regarding the 4th Amendment as technology has made the old reasonable expectation of privacy in Katz threadbare in terms of requiring warrants just as technology made the older Olmstead trespass standard less useful. See
http://thefederalist.com/2017/11/28/supreme-court-hear-important-fourth-amendment-case-generation/
One suggestion is that the Supreme Court adopt a mosaic approach that police must acquire a warrant if their information gathering reaches a certain level. Automated information systems in government and by private companies allow law enforcement to determine whether or not people are present in a home (computerized electrical and gas meters), analyze a complete network of communications of a person of interest, rapidly determine things like finances, travel, web searches, email headers, etc. by simply using their subpoena powers rather than seeking a warrant. Even bio-metrics that identity of people can be ascertained by accessing third party online photos and postings without your permission or knowledge. The justification is that we can more easily catch criminals that way by automating police work. Obviously, the counterpoint is that privacy is vanishing when every purchase, activity, and other records can be searched with a few mouse clicks. The joker in the deck is that essentially the law has become unknowable in many circumstances due to overly broad interpretations of statutes, triggering type statutes that rely on administrative regulations and agency decisions, and the overall mass of federal and state laws.