To some extent this is the NRA’s position. Responsible ownership has always been their position. Also, enforce the laws for those who are not responsible! Most states require training and a license to hunt or carry concealed. “Well” trained is harder to define.
However, since there is the 2nd amendment, firearms will not be like cars. Registration is a tricky concept when dealing with a constitutional right. It is easy to abuse the right when there is registration…. Insurance is another interesting and good idea for an individual (frequently instructors for firearms carry insurance in case of a problem), but probably can’t be enacted (required) for a constitutional right, either.
Yet, all that being done, would not stop the criminals or the terrorists from acquiring or using firearms. The truth is that there is little problem with the law-abiding citizens and firearm ownership. Since the law-abiding citizen is the only one that would go to the trouble to be licensed, trained, and insured there would be little to no impact on illegal uses of firearms.