How about donor organs? is there some infinite source of hearts and livers out there I'm unaware of?
Once you accept the fact that a given resource is rivalrous, you have to come up with some way to distribute it. Currently, that mechanism is money. If you get rid of that, then you have to replace it with _something_..whether it be algorithms, favor-economies, or central planning.
Now if you want to argue that insurance companies distort the market, then I probably agree with you (though I think there is room for interesting discussion here...). Certainly I think a single-payer solution is preferable to the current status quo, which mandates that individuals purchase insurance from private companies.
You can also increase the supply by opening up medical schools and giving more students the chance to attempt to become doctors without needing 10s of thousands of dollars.
Medical professionals and equipment are finite, tangible resources. No matter how capitalist or socialist your health care system, you are subject to limits on how much time, energy, and money you devote to health care.