For profit isn't necessarily inefficient and government isn't necessarily cheaper.
Here car insurance is a govt monopoly - everybody pays the same irrespective of what car you drive or how old you are. The only weighting is years of no-claims.
The result is that with max no-claims I pay >$1500 vs <$300 in the UK. Partly this subsidizes 17year olds being given Ferraris on their birthday but mostly it subsidizes the large "transfer payments" the government makes from the coffers of the insurance corporation to the treasury.