Earlier quoted context omitted.
It's worth noting that Kaiser is a nonprofit that is both "payer" (insurance company) and "provider" (hospitals and clinics) in the US - so I would argue that they have exactly the right incentives that you outline - maximize patient outcomes for minimal cost. So I don't think you necessarily need the government involved.
This begs the question: Why are for-profits winning over Kaiser, Why isn't the Kaiser model everywhere ?
Why do we have for-profit vs non-profit healthcare providers in the US and does that affect how they're run? The main difference is that non-profits tend to be more profitable since they don't pay taxes, but otherwise they're pretty much the same including the high bills. They are required provide some threshold of charity care, but the forms you need to fill out differ from provider to provider.
The 2nd question is why aren't there more dual payer-provider systems? There are two big barriers, firstly you need a large amount of capital to start up either a provider or a payer system, to do both you need even more money. Secondly you need expertise in running both a provider and a payer. Over the past decade a number of providers tried adding insurance arms. Many failed. In NY, both Northwell Healthcare and Crystal Run's insurance arms were started and closed (and these are both well-run healthcare systems) [1]. What's been going on recently in healthcare is that insurance companies have been starting joint ventures with providers. Even Kaiser has struggled when expanding in new markets, especially on the east coast (they're mainly on the West Coast where they were started as a way to provide healthcare to workers at Kaiser enterprises)
[1] https://www.recordonline.com/news/20190329/crystal-run-to-pu...