Irresponsibility is the cause of an action that could end up in jail. insurance doesn't keep you from killing someone it is taking money from people who can't afford to pay to have what the Second Amendment says they are entitled to as American citizens. The cost of insurance will infringe on people's rights therefore it will not pass as we see in NY it was just shot down.
They lobbied to make sure seat belts were mandatory. They did crash test dummy tests and began rating cars on safety.
They put pressure on the car manufacturers to have air bags in their cars (to improve their safety rating).
Car manufacturers have come up with many other features to make cars safer knowing that doing so will help sell the cars.
So then let's ignore the entire benefit of victims of a shooting event being compensated by the insurance company and focus on how an insurance company will reduce people getting killed.
1. You will have to show proof of insurance to buy a gun and ammunition. In this way the insurance company can be notified of every purchase you make. These are huge companies with powerful computers and algorithms that could immediately notify them of a red flag before you even leave the store.
2. In order to reduce your insurance cost you could provide more information than would be required, having that extra information like access to a mental health professional, and access to their social media could help the insurance company spot a potential mass murderer or other terrible event before it happens.
3. One of the biggest problems with gang violence is that the gun information is not in a computerized registry, so when they find bullets and shell casings they can use that as evidence in a trial, but it doesn't help them find the gun. If every gun in America had to be insured from the moment it was manufactured all the police would have to do is send the information to all the insurance companies when a crime is committed and they could check their records and find the gun. Yes, the gun will probably be listed as "stolen". But prior to being stolen it was sold at a retail outlet. If all the guns used in gang violence in Chicago can be traced to two or three retail outlets you can shut them down. Also they would have sold the guns to a "mule". A guy with a clean record who will then say the gun was stolen. If it turns out this guy bought 200 guns used in crimes, all of which he reported as being stolen you can convict him. So this will definitely make it much harder for criminals to get guns.
4. If you require insurance and then you catch a suspect with a gun it gives you a valid reason to hold the person for a day. Again, you can test the gun since it is now part of a crime and if it turns out that you have the bullets and shell casings in other criminal events you can charge this gang member before you would have had to release him on the streets.