> The hospital is required to take Medicare, and Medicare tends to be unwilling to pay the true cost

This is actually the reverse: the reason why Medicare is so much more cost-effective is that they're more consistent about paying only the true cost and have the marketshare to negotiate that up front rather than haggling over every claim. The hospitals favor private insurance because the margins are higher and they don't face the penalties for over-charging the US government if they get caught inflating prices.

How do you know that it is true cost? Given the number of inner city hospitals with serious financial trouble, I strongly suspect that they've gone below true cost!