The NHS is far from perfect.
But it is miles better than what a lot of Americans have, even when they've got health insurance (some of the companies have policies such as voiding insurance for cancer if you don't declare a mistaken diagnosis by a previous doctor, and many seem to have an attitude that makes our car insurance industry seem paragons of customer service when it comes to paying out for procedures*).
Just on that point, Insurance anywhere in the world is big on knowing what they are insuring and all insurance companies are the same, HMO's are the worst in the USA, but when you go to buy an over and some guy says, yeah this is the same as that oven over there, identical but its 1/4 the price you think, "hang on a minute" . People go for a supposedly good deal and get screwed often, better insurance companies screw you less.
As for paying out, car insurance, in all honesty is normally minor amounts compared to costs of health care which can run into the 100k's easily for severe/long term conditions, how many of the most frequent car crashes result in anything more than a couple grands damages.
As with any insurance company, the bigger the pay out the bigger the investigation and the harder it is to get the money.
Theres up's and downs for both systems, private healthcare there is about as good as it gets, NHS here misses lots of things and makes lots of mistakes but serves more people.
Frankly you can not have private healthcare quality for everyone, no country in the world can afford it, its impossible, same way no country can afford to give everyone a ferrari. People get paid, the healthcare industry is no different, it costs to treat people. Why does everyone expect the best free healthcare but dont' also demand free cars and free bigger houses, all 3 idea's are completely as unfeasable as each other.
It would be great if healthcare was easier, cost less and was more reliable across the board, its not, its expensive and mistakes get made. You want less mistakes and less risk you pay more.