Maybe it’s the legal regime dating back to WWII wage controls that’s enforced a system of third-party transactions tied to your employer?

The state of insurance undercuts the quality of the services, actually. It raises costs, changes treatment decisions, and has generally warped our system towards one focused on lucrative, worse-outcome behavior on the part of doctors and hospitals. Much of this related to the putting off of preventative or early treatment due to OOP costs; people gamble that their conditions won't worsen so that they won't be forced to go into debt for treatment (if they can even access credit). Many lose that bet and end up having to pay orders of magnitude more for late treatment.

Speaking personally, I had surgery for a long-standing condition, which the surgery failed to correct. Under a different system, my issue might not have become chronic, I might have been able to make a better decision regarding the surgeon and surgery (e.g., getting a second opinion, follow-up imaging, etc.), and I would not have missed a follow-up appointment after losing my insurance. Because I'm an American, living under this system, the circumstances of my treatment and subsequent outcomes were drastically shifted towards unsatisfactory.

Actually, compared to OECD countries, the quality is pretty bad.

The US is at or very near the worst among OECD countries in: life expectancy at birth, healthy life expectancy, rate of obesity, disability-adjusted life years, doctors per 1000 people, deaths from treatable conditions, rate of mental health disorders, rate of drug abuse, rate of prescription drug use.

It would seem health insurance companies charge a lot of money for not such great healthcare.

Those measurements you cite are not direct measures of the quality of healthcare services. Many people have poor health outcomes because they cannot afford to receive services from a healthcare provider. People often wait until their health situation gets out of control, because they are afraid to incur burdensome medical bills.

> I don't think anyone doubts that American healthcare is technically of good quality

Good healthcare is available to some in America, but not all healthcare in America is good.

> Its the insurance system that is broken.

Financing is the root cause of some of the other problems, but there are absolutely problems with care quality, too, even if resolving financing issues would lead to those going away.