Driving licenses are "a seal of approval of your driving prowess" in many countries other than the United States; American drivers frequently can't drive for shit. (Sorry, but I live in the UK, with approximately half the per-capita adjusted road death/injury rate to the US, and a driving test that's notoriously hard.)
As for "not that much death and dismemberment" because of the easy availability of firearms in the US, it's noteworthy that the level in question is a couple of orders of magnitude higher than in the UK, where firearms ownership is rare and tightly licensed. I wouldn't argue for a total ban -- if nothing else, North America is full of interesting and exciting wildlife, to which many people live in close proximity -- but there's no obvious need for city dwellers to own semi-auto rifles and handguns, and requiring those who do to carry third-party insurance in case an accidental discharge ends up injuring someone is an absolute minimum.
The thing here is that accidental discharges don't happen if you properly maintain your weapons and ammunition, and if you handle them properly.
As an example, I wouldn't store any weapons with a round chambered, wouldn't store weapons loaded, and wouldn't use any rifle or pistol rounds in an apartment for self-defense because of over-penetration concerns. The case where you have an accidental discharge and it hurts someone or something is entirely preventable using common sense, and so I don't believe that we should require insurance against what is honestly improper and unsafe tool usage--that burden should rest on the person who caused the accident.
(On the other hand, if you're as sensible as you say, I have three words for you: no claims discount.)
http://www.locktonrisk.com/nrains/excess.htm
Quick searching is not instructive as to pricing, but I guess some homeowners policies cover gun accidents, and such insurance is not particularly expensive.
I'll just mention ratios, that I'm a USAA customer who's father was in the military (eyesight kept me out, but they have a special category for us because we're better risks), that I don't have very many guns, have a monitored alarm, and no longer get the no claims discount (http://en.wikipedia.org/wiki/2011_Joplin_tornado).
All that said, I pay 4 times as much for the extra protection needed to cover loss of my guns as I do for the liability insurance that's an enumerated part of my renter's policy, and the single instance liability they're on the hook for is 35 times greater than a total loss of my listed guns and scopes.
I don't think it's merely a solution in search of a problem because we seem to spend an enourmous amount of political energy on the topic in the way we don't on 'who gets to own a Ferrari'. Ferrari owners pay the exorbitant insurance fees for the privilege. You're a firearms enthusiast who would like to own a .303 British? More power to you. Pay up.
This isn't possible if one side's position is 'I get to have whatever I want, because'. You can't buy a thing that plugs into your wall socket that isn't UL certified, what exactly, is so uniquely special about firearms, of all things.
One wonders, why? At least in the case of licenses and insurance, we see folks all the time (anecdotally illegal aliens doing work, though I'm curious how well statistics actually back that supposition up) drive with neither, usually to no ill effect.
Insurance usually doesn't seem to help very much in the case of a totaled car, and mostly just seems to be a captive market for insurance providers. Drivers licenses are more for glorified identification than a seal of approval of your driving prowess--look at the highways near any major city.
And yet, here we are, with millions of firearms owned and honestly not that much death and dismemberment because of it, all without licensing and insurance.
We don't need a compromise--it's a solution in search of a problem.