They would want you to pay for as long as possible so they would want you healthy.

Anyway as someone else pointed out it's just a roundabout way of saying they want universal healthcare, as that's exactly how it works, you pay a "subscription" through your taxes, get access to all hospitals for free and they have the incentive to keep you healthy because that's how they guarantee people are there to pay taxes. Yes it's pretty much the same thing as private insurance, without the "for profit" part and with a monopoly in the hands of an entity that doesn't have the goal is maximising profits so they can force companies to sell their services for the price they're willing to pay.

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.