Hospitals by their very nature are anti-competitive. There is no way that you as a patient can shop around. You do not have the necessary knowledge to compare the treatments, evaluate the options and decide on what represents the best value.
Of course, as you point out, it is very difficult to evaluate medical professionals from a layman's perspective. I ended up talking to surgical nurses in the hospitals the surgeons worked out of. It was not easy finding the nurses, it took persistence and physically going to the hospitals in question and asking real nurses which nurses I need to talk to. Then I had to track those nurses down, and convince them to talk openly. It helped that I have a cousin that is a nurse, and she gave me some contacts / introductions.
I was glad I did it though. One of the surgeons on my shortlist had a very poor reputation amongst nurses as to his skills in the operating theatre, so I managed to avoid him.
As an aside, considering just how critical the nursing staff where about this guy's skills, it is incredible to me that he is still allowed to practice...
Moreover, even if NOT ALL patients shop around, say, even just 10%, there would be enough momentum going to put pressure on doctors to improve quality of care and decrease costs.
Plus, hospitals are not in habit of committing to agreed upon price in advance. So, even if you shop around the amount they will charge you still can be much different then the one you calculated.