Wouldn’t competing hospitals and physicians still put downward pressure on prices just like any other good or service?
Your example can be said of food. Everybody needs food. Essentially a farmer could withhold his supply and jack up the prices since the consumers absolutely need food to survive. But all that does is drive those consumers to other food producers who charge less just as a patient/consumer would be driven to another physician or hospital.
The past 30 years have seen inflationary prices in the medical industry but not because of the reasoning you’re pointing out. Could it not be traced to the distortions caused by third party intrusions (insurance/government regulations) and/or monetary inflation?
What was free market medical care like before government got so involved in the industry?