Time isn't a factor, if you have insurance that covers it you rarely if ever know how much it will cost.
https://www.cms.gov/priorities/key-initiatives/hospital-pric...
Almost no medical practices in the US will tell you what the price is until after you've had the service there, and that price might even vary across multiple trips depending on how your insurance processed the claim _that_ time. (Ask me how I know.)
So you would need to schedule appointments and have the entire medical procedure done multiple times to have any price visibility, and even then, there's lots of nonsense around things like doctors operating in hospitals but billing separately, which completely obfuscates the end cost to someone.
People can and will choose the cheaper option, but I personally, as someone who has limited time, cannot afford to go through totalling all the bills for having the same procedure done twice every time, with all the months of back and forth for setting up being a new patient, scheduling things, being lied to about the doctor being available for the consult, showing up and having the staff gaslighting everyone about how it's not a problem on their end, everyone angrily waiting in the waiting room clearly got their appointments wrong, even if you have the reminders written by the staff at your pre-actually-seeing-the-doctor visit in hand showing their system is wrong...
It's incredibly exhausting and involved to visit a new doctor for even the most trivial things. Most people cannot afford the time commitment to do that multiple times even if the procedure is something where it's not a one-off or urgent.
The US seems to have some sort of racket going on in healthcare. What I've heard about their regulation makes sense if interpreted as a fairly basic system built by lobbyists to extract money from people, using emotional blackmail to break up political opposition. Otherwise it seems a bit silly.
Making it more transparent won't change it.
"Joint replacements can't be a free market" is simply nonsense.
https://www.cms.gov/medical-bill-rights/help/guides/good-fai...
Health plans are now required to give comparison shopping tools to their plan members. Those can be difficult to use for patients who don't understand the system, but it is at least a step in the right direction.
Basically I think this is over-weighting the importance of emergency care. I don't have any stats on hand, but people entering the hospital system at ultra-short-notice is quite rare in my experience compared to having a bit of time to think. Most healthcare engagement seems to be for planned consultation, planned surgeries, emergencies but with hours to spare and then urgent emergencies. In about that order of frequency, and heavily weighted towards the first few cases.
2. People are more than capable of figuring out which of two options are cheaper. The ability of consumers to reject goods over tiny price differentials is legendary.
3. If people aren't up for the challenge of figuring out what to do in an emergency, then in a fair market you'd probably find the insurer starts to step in and assist. If the insurer has to compete for business, they have a very strong incentive to make sure their customers get good and cost-effective care.
4. And on food markets; those are distortions to the market, not limits on it's freedom. A free market can be distorted with subsidies and it is still free. The important factors are whether new providers can enter the market and whether they are allowed to compete on price and service quality.