To make sure that I'm following, is what you are saying that the risk is that there's a big enough disaster that the insurance company(s) can't pay everything that's owed to those who have been harmed?
Correct. I'm saying that private insurance is good at insuring uncorrelated risks, like car wrecks, but usually, only the federal government can enable insuring correlated risks. Suggesting that mandatory liability insurance can a good policy alternative to regulation is correct, but if you are regulating something large enough with correlated risks, only the federal government can every really provide insurance. Banks pay FDIC insurance, and bank failures are usually covered by this, but it's pretty clear that the pool of money to save banks is backstopped by the federal government in case of a cascading failure.
Even for uncorrelated risk, if the payout event is bad enough, it can't be insured. For example, insurance on a nuclear plant is backstopped by the federal government, because no private insurer could ever really insure the worst case scenario.
