Car insurance is also a mandate. It works well. The healthcare situation isn't because it's a mandate, it's because of inelastic demand. A situation that's not comparable to gun ownership.
Arguably, in cities without good public transportation, car insurance has inelastic demand too. Further, it's being used to do things like backdoor in ubiquitous surveillance using GPS.
One also wonders what the effect car insurance has had on repair costs, for similar reasons as healthcare.
My point is that adding an insurance burden onto guns isn't a clear win, unless you assume that further government intervention (by the queerly circuitous route of private-sector insurance involvement the US seems to like) is unquestionably good. It also doesn't make sense to require insure for guns that are used for target shooting a few times a year.
Frankly, given the huge number of firearms and owners in this country and the rather small amount of violence they produce, it would probably be a bigger win to ask for mandatory insurance on pools, ladders, fast-food, sugary drinks, and so forth.
How much of the current fuckery of healthcare and costs is directly due to the insurance complex?
Hint: it's nontrivial, and in fact is a direct cause of rising costs in healthcare.