It certainly sounds simple. I don't know how far the governments control would or should reach in to the business but I suspect it would have to be an all-or-nothing approach, which means a complete and total reform. Considering the power that the insurance sector wields it's the sort of idea that could end a ministers career pdq if they even so much as suggested it.
But for the sake of optimism I will mull it over. Assuming the government not just calculated and collected premiums but kept all operations in house there could be massive difficulties with keeping legal representation fair and even. I think this would be possibly the biggest hurdle to one big happy insurance "company" not turning into a game of who can afford the best representation, as it arguably is currently. If you had to personally claim against this entity in lieu of having an insurance company doing it on your behalf (or arguing why they shouldn't pay out) then a structure would have to be put in place to ensure that one claim was dealt with as fairly as the next regardless of the means of the claimant. There would be many, many other issues to address and procedures to restructure but my mind's already boggling.
As to who would run this huge abomination of a department, well who else than the people who know best? Those already running and working for existing insurance and underwriting companies. This is the problem with a reform, as I see it, it could realistically be little more than a change of ownership with a new sign and a lick of paint here and there to show that the management has changed. A wholesale "replacement" of the exiting infrastructure sounds good to me but the practicalities of it are overwhelming. I like to dream though because I can't see how a transparent, open and publicly accountable compensation service (what else would you call it?) could make any worse the hell that those who simply want their losses reimbursed by those accountable go through. I do appreciate that a capitalist economy does breed competition and that then encourages improvements in service, affordability etc. However it also incites the amoral pursuit of profit over the proper and humane treatment of victims of others negligence whether it be the result of a car crash or a building fire.
Please note that I'm not an economist, legally trained or even further educated. I'm a manual labourer with a couple of GCSEs, a few bottles of beer and a partner who I will have to support for the rest our lives as the result of a non-fault RTA, an incompetent solicitor and two corrupt expert witnesses (allegedly, we're still working on that). You can write me off as bitter or biased I don't really care because I know that poor people get a rubbish deal when they try to take on big business.