The reason no one mentions the Danish health care system, is because it is irrelevant to the discussion. Denmark is a very small Scandinavian country, with a population of around 5 million. About 90% of the population is Danish, the other 10% are an assortment of Greeks, Germans, Brits, etc. They have a very small number of true immigrants, and virtually no illegal immigrants. This means, most of the people of Denmark know each other, are familiar with each others families, and they live in small, tight-knit communities, for the most part. With the exception of Copenhagen, the country is largely rural. It's also very cold in Denmark, there is not as much 'hustle and bustle' going on, people tend to stay at home.
Comparing ANYTHING from Denmark with that of the US, is like comparing Mayberry to New York. Can you comprehend the silliness of such comparisons? Systems which may work efficiently and sufficiently for a small town, are wholly inadequate for a large metropolitan city. Cooperative arrangements which work well among friends and neighbors, don't always translate to large diverse groups of total strangers. It is simple human nature, and a matter of trust between people, but small communities tend to be more personally responsible.
Here is a good example of what I mean... My family moved to a small rural town when I was 10 years old, and basically, everyone knew each other. Our town had a Community Center. This building was free for the citizens to use at any time, it always remained unlocked and available for wedding receptions and meetings, and the only obligation was to clean up after yourself when you were done. Now this arrangement worked in a small town, with a small group of people who knew and trusted one another.... but such a facility was impossible to have in a large metropolitan city, vandals would have seen to that.
Much is the same with regard to nationalized health care. IF we lived in a country where 90% of us were locals who knew and trusted each other, a nationalized system may work, especially if our populous were small and isolated due to the cold climate. But that is not what we have in America. Instead of 5 million with 90% natives, we are 300 million, with less than 2% natives. Instead of living in a culture where we may travel 30 miles in our lifetime, we live in a hustle-bustle society constantly on the move and on the go. It is because of this stark difference in culture, population, and condition, that comparisons are irrelevant.