It's not really backwards though.

There is nothing to stop a non-profit hospital from being as efficient as a for-profit one.

In that situation the non-profit will always be less expensive as the for-profit one unless the investors are willing to take a 0% return on their equity.

When it comes to what the market is willing to pay, what will a man that is about to die be willing to pay to be cured? Quite a lot I guess so there is nothing really to stop price gauging if the patient is paying for it directly.