As for your other point, I have family that handle insurance negotiations for hospitals. That is EXACTLY how every insurance company works, according to them.
Maybe it's a regional thing? I am friends with a VP of an insurance company, and have worked providing analytics software to insurance companies. The insurance companies I've worked with/know about are stuck between population health mandates and the providers - and the people I know are all trying to really help people while dealing with the madness that is unreasonable expectations from the population (we don't want to pay anything) and the providers (sorry we are trying to rent seek as hard as we can!) - I would not wish the responsibility of being an insurance provider on my worst enemies.
This here seems to be the point where everyone is talking past each other. The key part of a national health service isn't
just the insurance / payer side of things. That's half the equation. The other half is that those places have publicly owned
hospitals. for example, UK's NHS isn't actually an "insurance scheme" the way Americans would understand it, it's a network of publicly funded medical facilities, which guarantee equitable access to medical care for all citizens.
So both the providers and the payer are public. if you have public money but private providers, you just get profit-gouging: the industry reshapes itself to work out how to extract the maximum limit of public money, while providing the least actual service. That's why you get medicines that "cost" $200 on American Medicare, but only $15 if you buy them at a pharmacy. When people question the "costs" they get told by the providers not to worry about it: after all, it's the "governments" money, not "your money", right
?
The provides have worked out pricing schemes to guarantee they extract that exact limit of medicare benefits each person is entitled to, regardless of how much treatment they actually need. That's the equation they've optimized: extract the exact maximum amount of benefit money possible. If you double the medicare allowances, they just basically double all prices, since the amount of treatment people needed didn't really change, but they still want to optimize to extract the maximum amount of money. That's why it's a broken system.
Public money + private providers = corruption. That's the reason America's system is so costly. First, you need well-run state hospitals. If you have state hospitals, then at least any profits are going to get spent on infrastructure.
So you're right single payer
alone won't fix the problem, because it's private hospitals with no regulation on prices combined with a "feeding frenzy" to get the most share of the public money that's mainly responsible.
In fact, this identical structural problem plagues a
ton of things in the USA. Since you have so many private colleges, and you have publicly funded student loans, there's a similar distortion and massive cost blowout in the student loan sphere, because private colleges set their fees
based on how much student loan money they can extract from each student: that is their entire business model: extracting the maximum amount of government money per student, not providing a decent education.
The take-home lesson here is that you can't just have governments throwing money into a private sector and expect the amount of service to magically increase: the private service providers just ramp up prices to match the influx of cash until equilibrium is reached again. You need the government to build actual state-owned infrastructure with that money, and
compete with the private sector. That's how social democratic societies actually make things better: state-owned enterprises compete against private-owned enterprises thus keeping costs low. It's normal supply/demand economics.
Pumping taxpayers money into the private sector isn't "socialism" at all, it's actually
crony-capitalism:
https://en.wikipedia.org/wiki/Crony_capitalism