1.You mean they incur in the exact same kind of legal practices as private groups, with the same frequency? Given the difference in position, methodolgy and resourses, I doubt it, but I don't have any evidence pointing to either side about the behavior of Universal Health Coverage systems. I'd need time to ask a few people and find a few sources.

2.I don't think it's a matter of "layers" so much as one of how those layers are organized. The exact same amount of people can have productivity outputs that are radically different in function of the algorythms used to organize their work. Your post seems to imply that State services have more bureaucratic layers than public ones. I'd think that'd be something to decide case by case, but I wouldn't say it's a foregone conclusion: private insurances are infamous for being bureauratic hells too. Ones deliberately designed to mislead and confuse unhappy clients, at that.

Human rationality can be trained and improved, it's not an innate feature. To do that is part of the entire point of this site.

I hope you enjoy it. It is very interesting. Beware of generalizing from fictional evidence... but fiction is sometimes all we have to explore certain hypotheticals...

How different are the ways a society would treat citizens and various other people not covered by a system, such as Americans? What about tourists?

Isn't it true that Great Britain could provide better medical care if it diverted resources currently spent elsewhere? How are any other government expenditures and fungible things (like autonomy) ever justified if health could be improved with more of a focus on it?

Do you primarily value a right to medical care, or instead optimal health outcomes?

An intuition pump: What if a genie offered to, for free, provide medical care to all people in a society equivalent to that the American President gets, and a second genie, much better at medical care, offered even better average health outcomes for all people, with the caveat that he would randomly deny patients care (every patient would still have a better chance under the second genie, until the patient was rejected, of course). Both conditional on no other health care in the society, especially not for those denied care by the second genie. Which genie would you choose for the society? Under the first, health outcomes would be good and everyone would have a right to health care, under the second, health outcomes would be even better for every type of patient, but there would be no right to care and some people with curable diseases would be left to die.

If you would choose the first genie, your choice increases net suffering every person can expect.

If you would choose the second genie, then you're making a prosaic claim about the efficiency of systems rather than a novel moral point about rights for disadvantaged people - a claim that must be vulnerable to evidence and can't rightly be part of your utility function.

What if there were a third genie much like the one you chose, except the third genie could provide even better care to rich people. Would you prefer the third genie and the resulting inequality?