But otherwise I could be made responsible to pay for that other poor soul, the concept of risk distribution is too complicated and strokes of fate never touch me.
The issue in the US seems to be that even if you have a decent insurance plan you can still easily end up getting treatment by someone out of network. An anaesthesiologist or similar.
I wonder how it works if you're wearing a shirt that says "Don't touch me unless you are in network with xyz insurance plan #".
That won't save you. The doctor left the room and did a quick phone consult with a radiologist about your x-ray. Radiologist is out of network, so you'll get a big, fat, separate bill from them.