I have two active kids and am reasonably active myself. We have probably received medical treatment a dozen times in the last year, a combination of walk-in clinics, personal physicians and emergency room visits, depending on the circumstances. During these visits, the staff have used equipment, given procedures and prescribed medication and have rarely, if ever, inquired about my thoughts on treatment.
When my son jumped off the couch and cut the inside of his mouth open, I wasn't asked my opinion on treatment. And if I had been asked, I would have simply said "you're the physician, do what will make him better the fastest." The same would have gone for when I had pneumonia or had to get a tetanus shot after getting fishhooked while surfing. My goal is simple -- recover the best the fastest.
How does one legitimately comparison shop for a baby delivery? Imagine, for instance, Hospital A says their births average $20,000 and Hospital B says their births average $30,000. Which rational prospective parent chooses the $20,000 option?
Hospital care is not like buying a car, where most people can consider themselves educated enough to make a decision based on competing factors -- perhaps I'm willing to give up some handling for a lower price or I'm willing to sacrifice fuel economy for more power.
The average person knows nada about the medical practicalities of giving birth (or treating pneumonia, or even stitching gums) so they cannot make informed decisions about what are reasonable tradeoffs.
As other people have said, medical treatment is an inelastic need. Parents don't say "you know, I'm cool with 50% less effective treatment for my child at 50% of the cost" and people don't say "you know, 90% functionality of my leg will be just fine if you can bring this in at a discount."
So not only do you have a lack of informed customers, you have a lack of choice in outcomes. A free market solution for medicine is a fairy tale spun by people of certain social and political persuasions.