I'm not certain that this is a good analogy. In fact, the insurers' profit motive largely explains what happened. They leaned on government to mandate auto insurance, thereby increasing demand for their product. Then, they reneged on the promise they used entice the government to take action, citing yet another excuse for their high rates. This is precisely why I think insurers should not be part of the solution -- they are in it solely to make money for themselves, not to ensure that you or I have adequate health care.
I agree, but we are in a minority, even among democrats. The problem with insurance company competition is that they are competing for the healthiest. You can buy a good low cost non-group plan from a private insurer if you are perfectly healthy and young. They bet that you will not stay with them for long and most don't. They compete to cover large groups where a few large claims aren't going to break the bank, unlike for small groups. And they have lifetime caps so you or your premature infant probably won't cost them more than a million or two anyway.
They don't want the small employers. They don't want the chronically ill. They don't want the old. The model is a cherry picking model where insurance companies make a lot of money. A huge amount of money. The government ends up picking up the tab for the old and disabled. And many fall through cracks.