An NHS system, by its nature, has a fixed budget every year and has to control costs.
This is as laughable as anything you've ever said.
I can give you anecdotal counterpoints, my parents. They have great insurance, and they are pinged around to specialist after specialist for the most mundane ailments - to the limits of their insurance. There are duplication of tests ad infinitum, and my step-father is assuredly one who would NOT rate his care highly in any way, shape, or form. This is also a system which routinely charges $5 per Tylenol, which makes the Pentagon $25 claw hammer look frugal by comparison.
I can say I don't know in detail how the "overall satisfaction" metric is adduced. What we can say is, it excludes about 20% of our population though. Moreover, I'm sure the WHO accounted for this, but it is well known there is "medical monopoly power." People, even in the US, do not "shop around" for doctors or care. They pick one, and said doctor usually has to eat their baby before they consider changing (I know, suddenly every neoliberal on VN will tell us how they shop around rigorously for each and every medical ailment they have - against all the data compiled by the industry). So, even when an NHS system "limits choice" by assigning a General Practioner, this is actually what happens in a fully private system anyway.
being sure about anything from the WHO is the height of naivete or being in bed with their socialistic propaganda.
And finally, the bottom line to every spin doctored thesis:
single payer = better care for less money.
the bottom line can be there forever and still as wrong as two men humping.
If this were actually a market what would informed consumers buy?