The correct response is to both, within the confines, of course, of the Second Amendment. Even car owners with the safest driving records still have to maintain insurance.
As to whether what I propose "creates more problems", that's the purpose of having discussions like this. You've already gotten me to think about the virtues of fault-based vs. no-fault insurance. Contrary to what the usual suspects say, I do not have any hidden agenda to confiscate guns or to make it harder for the law-abiding to defend themselves or enjoy their hobbies. But gun violence is an epidemic. Stating flatly that registration and insurance can't address the problems that exist is unhelpful and myopic. Insurance is a time-honored and traditional way to allocate and mitigate risk.
No, the correct Constitutional response is to prosecute those criminal actors which survive and leave everyone else out of it. The Constitution does not allow for regulating the right to own a firearm, nor does it allow for anyone to lose "life, liberty or property" without due process of law, which includes being accused, tried and convicted of a crime. Writing new laws to make people guilty of something unnecessary which will not stop anyone from murdering someone in order to infringe their rights is liable to get someone unnecessarily killed, sooner or later, over something which should not have ever been an issue. In the meantime, though, the Constitution does allow for the arrest, trial, and prosecution of those who are out there performing acts of murder and violence, and those people can and do lose some of their Liberty as a result of either their actions or those of the court.
Passing another law when the Government cannot and will not enforce the very laws which it should be going by in order to stop lawbreakers which will only be created by the new law, not stopped by it, is a non-starter. I will crawl over broken glass naked in a blizzard to get to the polls to vote against anyone who votes for such nonsense, and will devote my resources to their electoral defeat.
My rifle IS my insurance from statist jerkoffs who think they can regulate it out of my hands.
"From my cold dead fingers" means something.
I'll be damned if I am going to have my rights, especially one which SHALL NOT BE INFRINGED nibbled away at. Stuff the insurance. I will own a rifle as long as I breathe. Take your insurance scheme and shove it. You would have honest people pay for the crimes of others. No effing way. End of discussion.