UnitedHealthcare operates in DC and all fifty states and has north of 26 million customers. Kaiser operates in eight states and DC and has around 12 million customers. It's safe to say plenty of folks use Kaiser when it's available.

The downside is that because Kaiser operates as an HMO, any specialists you need to see must be approved by your primary care physician (typically GP, ob-gyn, or uro). They have an abysmal record with mental health, and folks who need chronic non-routine care often struggle to get that care. If all you need are routine checkups, cheap prescriptions, routine immunizations, etc. they're quite competent.

  Who specifically sets the prices?

Kaiser is a vertically integrated HMO. Like other HMOs, to get coverage (with a few exceptions) you need to see a provider within the (Kaiser) network. As a vertically integrated HMO the providers in the Kaiser network are all employed directly by Kaiser. So Kaiser sets the its own employee compensation.

It's worth noting that all insurance companies are required to emergency care at any provider as if it were in network. If a Kaiser member were to go to a non-Kaiser ER they would see the same cost as if they'd gone to a Kaiser ER. Whatever Kaiser pays to the out of network hospital/staff is almost certainly negotiated beforehand.

So, I don't disagree that there are inefficiencies with private payers, but I do disagree that they're significant, or the reason US costs are so high, or that insurers deny so many services. You can see this for yourself with Medicare's admin overhead. Admin overhead is, roughly, the ratio of money spent by insurers to money insurers pay to providers. Medicare has "low" admin overhead --- but that's in large part because they serve the most demanding segment of the market. If Medicare covered 30 year olds, their admin overhead would mathematically be significantly higher: same money in, much less money out.

I agree with you about the efficiency of HMOs, but customers hate HMOs.

A useful Google search: "National Health Expenditures by Type of Expenditure and Program: Calendar Year 2022". It's a single spreadsheet, and it's really something. It covers insurers (public and private), providers (hospitals and outpatient), facilities, state health care programs, even dental, all on one sheet. The numbers are hard to get around.