Medicare is better than private insurers at regulating prices (to wit: when Anthem tried to adopt Medicare's rules for anesthesiology comp, they were excoriated, and the governor of New York proposed a statute preventing them from doing it). But they still wildly overpay for services; in fact, what they pay is not really materially different from what private insurers do (it's less, but not by that much).
to wit: when Anthem tried to adopt Medicare's rules for anesthesiology
comp, they were excoriated
How does that work? Do the providers have cartels that set prices?
Unless you're Kaiser, in which case you're the insurer and most of the providers, which is the whole idea behind Kaiser. But every other insurer --- and overwhelmingly, Americans aren't on Kaiser --- is beholden to providers. And of course, Kaiser competes with providers for service providers and vendors.
Providers is also kind of vague. You show up at a hospital, does the nurse practitioner who does your intake set the price of your ER service? Who specifically sets the prices?
HMOs, for all their problems, have many advantages as well, such as the aligned incentives you allude to.