I would like to explore this claim.
Let us start with use. How does registration of a firearm enforce responsible use of a firearm as compared to an unregistered firearm? What has changed impacting use?
The efficacy of registration upon the safe use of firearms is influenced in part by whether (as is the case with motor vehicles) it is coupled with an insurance requirement. If we assume that is not the case, then to me the primary value of registration is that it links a firearm to the person who is legally responsible for its transfer and disposition. That affects usage in that the legal owner will likely take care to keep the firearm secured against theft, and will inventory his firearms to detect theft and unauthorized usage. When a firearm is used in a crime, it can traced back to its owner. At that point, other aspects of the law determine whether the legal owner is responsible for the harm caused. If he has documented the transfer of his firearm, then he is off the legal hook. If he has reported the theft, then he is (IMO) off the legal hook.
Obviously, the state of a local jurisdiction's rules on liability for unauthorized use will play a role. In the context of motor vehicles, some states do not hold the owner responsible for unauthorized use of his car, but in other states, in certain circumstances, the owner is deemed responsible especially if has acted irresponsibly (such as leaving the keys in the ignition of an unlocked car). If linked with an insurance regime, victims of gun violence (just as the victims of motor vehicle use) can be compensated for medical costs under a no-fault system funded by premiums paid from a broadly-funded pool.
The liability regime for motor vehicles is obviously different than that for firearms, and will vary from place to place. But registration is the tool by which the deadly device (car or gun) is linked to the person with color of legal responsibility, and effectively compels him to dispose of the device in a documented manner.
In short, registration is consistent with reasonable notions of owner responsibility. Confiscation is not the objective, and the Constitution's protections are the means to ensure that.