In general principle, I agree. There will be numerous cases however, where that's neither pragmatic nor possible. Namely, I'm thinking of injury claims arising from personal or commercial auto, general liability or anywhere else the injuries were sustained as a result of another party's negligence. If you go to a store that has an improperly maintained staircase which gives way while you're on it and you break your ankle, you're not going to want to, and may not even be able to, pay that bill out of pocket, and it's reasonable for you to expect that store (or by extension their carrier) to cover your medical bills.
Good exceptions. I figured there would be some, hence my ‘most.’ I hadn’t thought about your scenarios. There is potential for things to get unavoidably messy when there is a potentially adversarial relationship between ‘patients,’ ‘doctors’ and payors. I’d like to believe that in the majority of health care interactions patients and doctors are trying to be on the same side. My ‘general principle,’ where applicable, would save everyone, except the third party parasites, a lot of money.