In Spain, for example, we have a private system but it is extremely inefficient in some areas (and very good in others). Of course, you can have private insurance, but you still have to pay your social security. Curiously, the only ones who can decide which system they want are the public servants...
You cannot really avoid the fundamental constraints - anywhere in the world, there are only so many doctors and so much money available for treatments. IDK if USA has a shortage of doctors, but plenty of European countries do. A country like Romania just cannot give its doctors big enough wages to stop them from seeking employment elsewhere, where they will get five to ten times as much (UK, Germany, Switzerland). As a result, local hospitals are seriously understaffed.
Where I live, having personal connections to good doctors gives you an advantage - you will be examined and treated faster. Then there is outright nepotism.
The outgroups are different than in America, but there are always people for whom the system sucks.
What you say may be somewhat true in the context of transmuting the US's "private" bureaucracy into bona fide "government". But it's certainly not a "fundamental constraint" that's impossible to solve. Rather it's a failure of organization, whether critiqued in terms of bottom-up market failure or top-down governance failure.
This is incorrect. Most of the paperwork is done by administrative staff. Paying for that giant staff + the actual medical professionals is why things are so expensive.
Hospitals are not stupid, they won’t waste their most valuable resource (healthcare time) on bureaucratic paperwork.
X is the amount of medical care available
Y is the amount of medical care wanted
If Y < X there is no problem with any of the systems. And, obviously, a certain amount of inefficiency doesn't affect patient care. Plus, perhaps relevant today: when shit hits the fan we can scale up available care quickly.
If X > Y it doesn't matter which system you choose, someone will go without. You can change who goes without but you cannot fix the system by changing the method of dividing care.
Can things be somewhat improved with better organisation? Sure. Probably. But let's not overestimate it either. Let's take a dream scenario: optimal organisation can make 20% more care available. How much more care is wanted? I think we can safely say the US population wants 200% or more than the current system provides. Whilst nobody's opposed to improving organisations, it cannot fix the problem.
Fixing the problem is something you can only do by doubling the medical training available. That'll be a lot of extra dollars, none of which go anywhere near patient care for at least 10 years, so I would expect a lot of strong opposition from a lot of sides. But it's the only way to fix things.