Health insurance generally has a fixed profit margin (state legislation). They have little incentive to reduce cost because then the entire pie shrinks. A nice example of where well meaning legislation can completely backfire.

Of course, passing costs to all insurance companies is really the same as passing it to all people paying insurance premiums, at which point you can just use tax money to get the same effect. At which point, it's probably easier to regulate it and have the cost passed to everyone buying a car.