I take resource constrained over purposefully obtuse system. I can understand that people and medicine is expensive and inefficient(compared to farming and factories or tech). But at least these public systems seem to attempt to do stuff. And not be piled with bureaucracy to avoid paying out or increasing the prices.

An important metric in efficiency of systems is overhead. For health system, it would be how much money is spent on purposes other than the stated purpose of the system. Private insurance is never going to optimise for that, because low overhead means high utilisation of funds paid in, and low profits.

Commonly complained about polish NFZ (which, unlike NHS, operates through contracting to semi-private and private providers), has overhead IIRC of around 1%.

I don't have links on hand (maybe someone more recently versed on US insurance can check?), but I heard of common overhead levels in private insurers in USA being in range of 30%, mainly for claim acceptors/adjusters.

I believe insurers are required to spend 85% of premiums directly on medical care: https://news.ycombinator.com/item?id=37977842.

(Not saying that's the right number, health care in the US is out of control).

Indeed. The incentives are all in the wrong place in the American system.