You might know that you won't have to pay more than $X, but you generally can't find out beforehand the theoretical cost of the procedure as billed to the insurer. So if you're on the hook for any fraction of that (as in your second scenario) then you're very much in unknown territory.
There is also the question of how well you really understand the terms of your insurance policy, which can be extremely complex.
What gets me about the insurance billing is that besides being complex for me, it is also complex for the insurance company - people on the other side are just people, after all, not some infallible super-intelligent aliens. And the failures that come out of that are kind of hard to believe for people used to nationalized health care; my favorite example was a $15k bill I received months after a surgery that I called back to get more details, upon which the agent said something like "oh that's coded wrong" and after a few keyboard taps I owed $500. What if I had paid the original amount without asking? How many people do not ask? And the cynical question, how often do the "coding errors" happen the other way?
There is also the question of how well you really understand the terms of your insurance policy, which can be extremely complex.