It is definitely true that in the US the quality of your medical care depends on the quality of your insurance.
My healthcare there was provided through 2 Health Maintenance Organisations, each centred around a major university medical centre. Both were superb, with much better treatment than the NHS offers. In America, land of free choice, the HMO option was considered less desirable by many because you had to choose GPs and specialists from the network and use the (amazing) university hospitals. As DH and I were both university employees, the cost was nominal.
Both HMOs also treated a number of patients on both Medicaid (for the poor) and Medicare (the very good public option for the over 65s who have made sufficient contributions). Not all health systems are happy with these public patients, which is simply wrong IMO
Several friends needed out of network emergency treatment. This was provided by contract. One had to be airlifted home from Eastern Europe. I know he was happily surprised at how easy the financial side was.
The moral is that America leaves far too much to luck; it is immoral. A good HMO is much better than the NHS. (I write as someone who credits the NHS for saving the life of DH. I do not think they could move quickly enough against his aggressive cancer to do the same today) A bad one can be a logistical nightmare. Private insurance can be good or horrible. Medicare and to a lesser extent Medicaid work pretty well once you are plugged in to the system, but making connections can be a nightmare.
I think it is mostly the uninsured who are bankrupted by medical costs, but I am not sure. In any case it should not happen in a civilised society.
TL;DR American health care is a story of extremes, and it is not just about what you are paying. The best is much better than the NHS. The worst has no place in civilised society