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.