You add the layers of bureaucrats regardless of whether it is at the government level or the private insurance company level. Competition helps, but in the medical insurance industry the cabal of a few large corporations is very difficult to break in the free market. The risk profile cannot be compared to car insurance, it is much, much higher in frequency and at the cost of claim level. Small companies cannot become health insurance providers. All they can do is resell the policy and pony off the risk.
Look at the NHS of the UK or the Cuba model extolled by Michael Moore in Sicko. All of it lies. Watch The Un-American documentary on Amazon Prime to see for yourself what government run healthcare really is like! That is without the BS! Just real life examples if you want government running your healthcare? We will have the same in the US if you allow the government run our medical care! Look at Amtrak, look at the United State Postal Service as how bad government runs things. Atleast, private companies have a budget in place. Government will run deficits over and over figuring it is the US taxpayers will be paying for it so, who cares?