It's interesting that we just assume it makes sense for the patient to self assess what level of care they need.

I wonder if there are sensible reasons to not have a single point of contact for unscheduled care, or if it is just dumb inertia?

I know there are some hospitals that have provided urgent care type services at urgent care type prices in their emergency rooms, I haven't looked to see if it worked well or if they are still doing it.

Spelling out what I implied - my experience has been the exact opposite of how OP is insisting that it should be. I'm not interested in typing up a report of my various experiences with the medical industry and scrubbing it of the right amount of identifying information merely to refute some prognostication that's so abstract it's not even wrong.