That's the problem with a fragmented health system. A lot of doctors are specialists in a certain procedure and want to repeat their pet procedure on every patient they see. I saw this with my ex who had chronic leg pain. Most of the doctors we met just wanted to repeat what they do every day (and makes them good money). There are only a few doctors who have a wide knowledge of the area and can recommend a treatment that actually fits.
The only missing peace is the a doctor on the care team who chooses which type of surgery is best, becasue once one is referred to a specialist: they are going to get that specific specialized care.
Perhaps such a system incorporates some aspects of private enterprise, but ultimately it should still be seen as a non-private system.
A truly private healthcare system would allow anybody to practice any kind of medicine at any time, without specific requirements and restrictions being imposed by the state.
It would be up to each patient to decide whether or not he or she will deal with a particular practitioner or provider of services. (Of course, this can be complicated in the case of an incapacitated patient, without prior planning.)
As is typical with non-private/socialist/communist systems of any sort, where natural free market balancing forces such as consumer choice, competition, and reputation are limited or absent, we should expect to see the inefficiencies you described, including significant quality variation, bureaucratic games, and misaligned incentives.
Healthcare in America seems very non-private in nature to me. It may not be as overtly non-private as the healthcare systems in Canada or many European nations, but it certainly does not seem to be private, either.
Even today, organized healthcare beyond vaccines and antibiotics has only a marginal impact on lifespan and healthspan. It's way behind sanitation (clean drinking water, sewage systems, waste disposal) and lifestyle issues (substance abuse, obesity, lack of exercise).
It is important to understand that in the American private healthcare system, the doctor is the true “customer” of the hospital, in that, with the exception of services provided by hospital-employed physicians, the things the hospital itself can actually bill for have to be ordered by physicians in private practice who have patients admitted to the hospital.
There is a tremendous power imbalance between physicians and non-physician clinicians and a powerful incentive to cover up bad acts by bad doctors who perform a large number of profitable procedures.