Unfortunately, your interpretation of the issue is off. This is not a national health insurance vs. private health insurance issue because even in the UK and countries like it, people have the option of buying private health insurance, too. After all, the Gards did raise enough money for the treatment. UK citizens are just guaranteed basic coverage, as well.
This is a medical ethics issue. If an American parent ceased getting treatment for his/her child at an American hospital that was the standard treatment for that condition, as rare as it was, and decided to try some experimental treatment that has never been used before for that condition, which even the doctor said had an 11% top 56% chance of improving conditions for the child, there is the possibility that that parent could actually be charged with medical neglect. Those chances would increase even more, if the parent was a poor, single woman, especially if she was black or Native-American.
The hospital was dealing with a very difficult decision for them and the family, and in their view, the risks far outweighed any possible benefit that may or may not assist this child. The hospital also has to deal with liability issues that I am sure is little concern to the parents, who just wanted to give their child a fighting chance, but does affect them as an institution. That would also be an issue, if this were an American hospital, too.
Yes, the British government may have more jurisdiction with the courts backing them, because they have a health system that is predominantly state run, but let us not kid ourselves that American parents are not restricted to what they can and cannot do, either, when it comes to the medical care of their children.