I got a bit derailed, but your comments warranted examination and response.
If you want to make it constitutional then put the insurance premium on ammunition for the gun.
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I'm certain someone can account for presence of ammo or not in insurance premiums. I know you are aware, that "right to bear arms" translation you are using now is not the version the SCOTUS held until the 1970's. That was when the NRA effectively changed our legal view for that to be an absolute right, unencumbered by rules, and diminished the clause "well regulated militia" which is in the same sentence as the right to bear arms. I'm not even arguing over removing guns though...just forcing gun owners to be more responsible.
As you might guess, I tend to accept the NRA's view that the well-regulated militia clause isn't a limitation on the right to bear arms. However, I don't accept their application of the Second Amendment to the states through the 14th Amendment. As I've indicated before, I think a state has the right to ban handguns unless they limit it by their own constitutions or laws.
We put that insurance requirement on drivers when cars became ubiquitous and the damage they did when used carelessly became an issue for the other drivers. As of right now
per Americangunfact.com 38% of Americans over the age of 18 have a gun and the US has 393M guns in the marketplace. Compare that with
only 276M registered automobiles. Yes, we have >100M more guns that cars in homes. With Open Carry and expanded concealed carry laws the guns are even more prevelant. They are now outside of the homes. Guns are the most common murder weapon (by magnitudes), the most common suicide option and accidental deaths.
This site has unverified (by me) information on accidental gun accidents. I'm saying, that the US fascination with gun ownership has now crossed that same divide that we did with cars where the volume of them results in more injuries by their users so those users should be financially liable for the responsibility they choose to bear.
There are more guns than cars. However, I think you also have to consider use, exposure to the public, and likelihood of accidental injury. Obviously gun accidents happen, and every one of them is a tragedy, but auto injuries and fatalities are much, much more common even though there are fewer cars. For example, in 2018 there were 458 accidental gun deaths in the United States. There were about 36,000 traffic fatalities that same year.
The bottom line is that driving a car is just a hell of a lot more dangerous than carrying a gun, because it's far more likely to be lead to injury or death. It's much easier to accidentally cause a car wreck than to accidentally fire a gun.
Why do I not consider intentional injuries and deaths in this equation? Because liability insurance protects against negligence and gross negligence torts. It doesn't cover intentional acts.
I also can't speak for every state, but at least in Texas, a car doesn't have to be insured if it's not driving on the roads and therefore exposing people to harm. A car that's just sitting in your garage doesn't have to be insured. I don't see much need for a gun that's sitting in a safe to be insured. Could I be convinced to require it for those who get carry licenses? Perhaps. Should gun ranges have to carry insurance that covers the shooters inside? Absolutely. But do I think a guy who keeps his gun in his home and uses it only for home defense should have to carry a special liability insurance policy? No, and there's a good chance his homeowners policy would kick in if an accident occurred.
And of course, you can see where the skepticism is going to come from. States have the power to set insurance rates. Will the State of California set rates at something reasonable, or will it gouge gun owners just because it can? Gun owners aren't going to trust states like that not to make insurance premiums do to the right to bear arms what a poll tax did to the right to vote.