High population density in Denmark - Yes, Denmark has more population density than in the US. Yet universal healthcare works in Iceland, Greenland, Canada and Australia.
US average pop density: 35/km^2
Iceland: 3.2/km^2
Greenland: 0.028/km^2
Canada: 3.92/km^2
Australia: 2.8/km^2
I'm aware that I'm going by the population density of all of USA instead of state level. However, the lowest state pop densities are South Dakota (4.4), North Dakota (4.2), Montana (2.7), Wyoming (2.3) and Alaska (0.5). This means we're counting five states that are just in the ballpark of what has been proven possible in Canada and Australia if you want to talk about size. And just to remind you, Canada is the world's second largest country, Australia is the world's 6th largest country and Greenland is the world's 12th largest country, if you consider them independent from Denmark.
You can argue that Greenland's abysmal population density should make universal healthcare possible, and that Greenland's healthcare is only possible through Danish subsidies. But if you want that argument, your point about Danish population density making it possible would be incredibly problematic, since following that Denmark actually has a much larger area to cover. So, you know, Denmark's healthcare actually covers a much smaller population density in that regard. Denmark would be the world's 12th largest country with an abysmal population density if you want to make that argument - Denmark prior would be just 2% of its total size.
EDIT: This is how they do it btw.
http://mastermind-project.eu/partners/agency-for-health-and-prevention-greenland/
Diversity. Of these examples, Iceland isn't very diverse. Denmark could also be argued to not have much diversity. But regardless of what you hear in the news, Denmark does have 10% of its population stemming from immigration. This is not an insignificant number, even though the US have "just" 73% of people being white and a 17,6% Latin minority, depending on what you care about when measuring diversity. There are a bunch of other statistic that demonstrate, yes, it is true that the US has more diversity than Denmark.
But if you lived here, you would understand it's not that simple. To get the obvious out of the way, we've always have had issues with a significant German minority. But there's something in regards to our demographics that foreigners don't realize. Thing is, due to our island make up, Denmark has a lot of incredibly diffuse dialects. The difference between West Jute, North Jute, South Jute and "rigsdansk", the dominant dialect, is so large that these Jute accents are
completely incomprehensible to Danes from other areas. This doesn't include the dialects that are just difficult to understand rather than impossible. Anecdotically I can refer to my ex girlfriend who couldn't even understand her own grandfather, even though they saw each other all the time. This ties in with real political tensions between the capital and the provinces. This doesn't show up on Wikipedia because people simply identify as Danish due to a moderately succesful nationalist nationbuilding in the 1800s. But it does not mean that there aren't real differences between rural and urban Denmark, if anything regards to values. In fact the question about rural Denmark could easily be argued as ethnic problems. But they aren't normally because Danes just so happen to identify as Danes. The cosmopolitan, Atheist, socially democratic Denmark that tourists meet does not tell the whole picture of the nation; in fact, your point about tribalism very much applies here. Another point is that we're incredibly racist, yet neither native Danes or brown migrants believe the other shouldn't be allowed universal healthcare. To answer your prompt, we don't universally do things "nicely", infact, I'm not sure we deserve the reputation of being a "nice" country.
Also if you really want to ignore Denmark, the point about diverging ethnicities making universal healthcare impossible is just a fallacy seeing how many countries succesfully do this with diverging ethnicities.
Freedom of religion/press/assembly and its relation to culture. This baffles me. So the US can't have universal healthcare because you have these... Danish citizens don't have these rights? European citizens don't have these rights? Is it because diversity feeds into this, as if European countries aren't diverse and don't have ethnic issues? Have you ever picked up a history book, or heck, just a book about contemporary demographics?
USA is not the exception. The Mongols are.
OTOH I can understand the difficulties in funding depending on state/federal level, I don't have an answer prepared for that at this point. There is a real difference in regards to parliamentalism and constitutional function - Denmark's constitution is laughably ignored at this point. I'm not sure how it is a dealbreaker however. The countries granting universal healthcare are incredibly diverse and I have seen nothing innate about US administration that makes this innately impossible. Now, what's difficult is trying to get such a fundamental overhaul through the US political machine, but there's a large difference between that and managing the policy once it's already in place. You're free to try to explain this to me - in the other thread.