It makes fundamental sense that one should be able to get lifesaving, or at least stabilizing care in an emergency regardless of ability to pay(*). Coupled with this should be some sort of mechanism for the public at large, or the system as a whole to manage those costs. However, given the way medical care is funded in the US I think doing so would be a monumental challenge. Let's be honest, if there weren't hanky panky going on, then Dale Folwell wouldn't have gotten hundreds of pages from (I think it was) UNC Healthcare saying the price for procedure XYZ is ... redacted .. while they simultaneously refused a contract for the state employee health plan that was something like 205% of Medicaid / Medicare figures. In virtually no other industry does the consume get to know the price only after consuming the product and in the case of medical it meets the textbook definition of racketeering with parties operating in private collusion to fix the pricing. I think what we are seeing now is but another symptom of an industry that is in crisis.
* - Important distinction needs to be made between emergency, urgent, and primary care as you said earlier. You'll get at least this much if something happens to you in many foreign countries. I remember back when I was in middle school, every year the 8th grade would go to France over spring break (side note on that, one class got to attend easter mass at Notre Dame). I recall one year, a girl received an appendectomy courtesy of the French govt. Not how I would want to spend my vacation, but better than the alternative.