We are, yes. But the average American would be shocked at the "customer experience" the average patient gets from the NHS. No such thing as a free lunch etc.
And I think if you have private insurance (i.e: double paying), you'll probably be paying more than the average American. I don't know for sure though.
The per capita expenditure on Medicaid and Medicare in the US is substantially higher than the per capita expenditure on the NHS, but Medicare and Medicaid only cover about 20% of the US population.
On top of that UK private insurance is much cheaper because it is 'top up' - you're usually expected to see NHS GP's first etc. and use NHS emergency rooms.
As such while it is possible to end up paying more if you're very high income, very few people in the UK will pay more even with private insurance.
You also have to factor in the (very large) public expenditure on healthcare in the USA. Americans also pay for part of their healthcare through taxes.
I am not sure what you mean by “average American” but millions of people in this country rely on emergency room visits for their health care needs. I suspect the NHS provides a superior “patient” experience.
Yes, the portion of my taxes that goes to the NHS plus what my employer pays in private insurance is less than what Medicaid and Medicare costs per taxpayer in the US.