I don't know if that is a perfect comparison. I have been wondering for a long time why liability cover on US car insurance policies is so low compared with UK liability cover. Default cover at a pathetic 100,000USD instead of the 8,000,000GBP that my last UK policy had.
I recently spent some time reading web sites suggesting how to assess how much cover you need and they suggest it based on the value of your assets. So it strikes me that while UK auto insurance is designed to have a high enough liability cover amount to make sure that those you harm don't lose out (within reason, clearly a person who loses a leg loses out, but the value should at least cover their medical bills and given them some compensation given the loss of income), US insurance cover is intended to cover you as a driver against the risk of losing your assets to legal action on the part of those you harm.
So although clearly the payout on auto insurance does go to others, to then cover their costs while health insurance directly covers your costs, the actual goal of the insurance strikes me as protecting yourself in both cases - at least in the US.