there are other kinds of problems. You are in the hospital, you feel queasy after surgery, nurse asks if you would like medication for that. "how much is it?" If you even think to ask that, usually they will not know, and can't or will not find out for you. So you can't make a rational decision about whether to deal with the discomfort or not.

Similarly, doctors often won't know the cost difference between various medication options, and those costs are always extremely difficult to figure out since there are usually various coupons that last for various amounts of time, which if you are willing to sort through you can game the system a lot. But in the long run it would be better if those games didn't exist. Switching asthma meds as you run through coupon time limits isn't really the healthiest thing anyway.

Or you are hit by a car, an amublance shows up "what will the bill be? Oh I'll call a different service" no, you can't do that while you are bleeding out, nor will you have any way of knowing which hospital will have the best rates to fix you, you don't even know what is wrong yet!

One might conclude that private, free markets don't make sense in this domain at all, perhaps.