Additionally: they're optional procedures, ones for which there are often long-accepted reasonable substitutions, and you can take your time researching and comparison shopping. Given those conditions, you could reasonably expect a market of competing private providers to do pretty well at yielding reasonable prices for effective services.
(Absent those conditions -- anywhere that care and treatment are oriented around alleviating illness and suffering with any kind of immediate debilitating effects or urgency, really -- you might well expect that a market of competing private providers doesn't yield the same effects.)
In other words you could expect the market to improve virtually all medical care. The "urgent" stuff you describe where a person could not temporarily postpone our shop around is in the vast minority.
I’m curious why you think this. When last I looked, a very large percentage of medical care was urgent care; in several demographics, it was a large majority.
There are a variety of ways of measuring ‘how much’ medical care is given, which has a big impact on how things compare. Cost per capital, for example, is a fairly intuitive measurement and it skews heavily towards urgent care.
The study cited in this article suggests emergency care accounts for just 2% of all healthcare costs. There's some other ways it could be calculated, but it's definitely not getting out of the single digits.
(Absent those conditions -- anywhere that care and treatment are oriented around alleviating illness and suffering with any kind of immediate debilitating effects or urgency, really -- you might well expect that a market of competing private providers doesn't yield the same effects.)