If you’ve been in a crash, and your car gets towed to a bodyshop, how much do you argue about repairs? Sure you can move it, but you’d have to get a tow truck to take it around to other shops to get estimates.
What actually happens is that when you have car insurance, your insurance company has worked out a schedule of fees in advance with the body shop, just like for health insurance companies and doctors. Same deductibles, in fact.
For elective procedures that can be scheduled in advance, like cosmetic surgery, it is simple to shop around.
There is also the problem of continuity of care. Even if I could shop around for each illness, I might lose out if a trend gets missed. Not to mention that if you have something serious wrong with you, how long are you going to wait for appointments for n different doctors, and how much will you pay for extra tests? Plus, lots of people have to take time off from work, which costs them.