It seems you read what was on your mind rather than what I wrote
An HMO is
mostly just an insurance company the contracts for care.
That is not the same as the care and insurance coming from the same company.
I don't think I misread what you wrote. I was addressing the history of one failed attempt at a provider-controlled health plan. HMO's initially (1980's, excluding Kaiser) were often formed by large medical practices, like Mayo, Geisinger, Cleveland Clinic, etc. and university medical centers. Thus, they were provider-controlled insurance. Many of those HMO's have been sold since to insurance companies because of the economics. Those that remain independent are in somewhat protected environments.
One driving force for the change was how to pay for out of system care. If a patient was out of the service area and got care, the HMO paid essentially list price for that care. (It is another story about how list price got disconnected from actual reimbursement.) Thus, the HMO's needed to expand their service area to avoid loosing huge amounts of money on such care. That led to regionalization and eventual take over by insurance companies.
Second, the larger the HMO (now just an insurance company), the more effectively it could "negotiate" price concessions from providers, even ones that weren't enrolled.
Finally, the large HMOs could afford marketing to attract younger and more healthy patients. Even if one excludes out of system care, the economics of an HMO just don't work, unless there is a continuous supply of young, healthy enrollees.
So today, although the HMO conceptually had some attractive aspects, it has become just another type of insurance. One example of which is/are the Medicare HMO(s), which remain physician controlled in name only (NB: HMOs still have Medical Directors).
John