Isn't that what insurance is?

Currently, I'm not big on insurance. Like loans, it separates the consumer from the product. A consumer's reluctance to directly pay for a product is what drives down prices.

Also wouldn't it incentivize hospitals to kick you out even before you have healed?

Yeah, like I said, it's not without its issues. Perhaps, when then patient is admitted to the hospital, a contract is simultaneously created which is an agreement between the hospital and patient that they will go through the full treatment required to help them. If the patient is removed prematurely, then the patient can file a lawsuit against the hospital for breach of contract.