The devil's advocate in me will tell you the truth as only I see it, not as the up/down votes in this board of mostly libertarian social darwinist and would-be technocrats will indicate:
The reason Denmark's system (or Australia's, or Canada's or S. Korea's, etc.) works is simply this: They have mostly one type of people there, and sometimes have a little bitty number of other types of people there, and mostly that little bitty number of people keep to themselves and open up exotic restaurants and try really, really hard to assimilate to the dominant culture and pay their taxes; and if they can't completely be as successful as the dominant culture, they will leave well enough alone.
In contrast, Americans will look at the person sitting next to them, see them as lazy, and refuse to pay taxes to support who they think is taking all of it. That person next to them may be poor, black, old, sick, hispanic, have down syndrome, is a teacher or a cop or a union steel worker or liberal or English major or hippie... ad nauseum. Whatever the reason, that person looking over at that "other" person flat-out refuses to allow "my hard earned money" to go to whoever the hell that is sitting next to them. It just won't happen.
Another thing, our poor and elderly are heavily, and I mean absolutely, unmistakeably, and incorrigibly self-effacing, self-hating, and in flat out denial over their station in life. They are told through attack ads during elections, through cable news, and through the water cooler, that blood-thirsty communists are alive and well and have a donkey replacing their hammer/sickle emblem. They're gonna take your guns, your religion, your money, your children and your property, give it to white trash and Mexicans and Godless welfare queens who drive better cars than you on the way to the food stamp office, and you'll be left with nothing but their hospital bills that you now have to pay because they bought a flat screen and new spinning rims for their car.
No one will ever admit that the heterogeneous society that is America collectively feels this way, because we didn't have a raw sore like the Economic downturn in 2008 for us to really see how much hate we have for one another. No one cares about each other when the bill comes.
I do think there is some notion of relatedness that ties in and leads to the system generally being supported. I'm not myself ethnically Danish, so I'm not sure it's purely ethnicity, but it's... something (and ethnicity probably does play a role in encouraging cohesion, if not the sole one).
Even in extreme cases, there's this general feeling of: yeah, so that person is an irresponsible alcoholic, but he's your irresponsible alcoholic, so you can't exactly let him die in the street. You've got to give him some basic shelter/food. I've run into that in other countries also, but only from closely related people: southern Europeans are very family-oriented, so if someone in their family is in danger of being a homeless alcoholic, they'll find a room for the person to sleep in, etc., since it's considered the family's responsibility to do something about them. Denmark doesn't emphasize family ties as much, though. Instead things tend to be done on a wider scale: it's sort of everyone's responsibility to take care of the proverbial crazy old uncle, rather than laying the responsibility on the handful of people who happen to be his nephews/nieces through accident of birth.
The reason Denmark's system (or Australia's, or Canada's or S. Korea's, etc.) works is simply this: They have mostly one type of people there, and sometimes have a little bitty number of other types of people there, and mostly that little bitty number of people keep to themselves and open up exotic restaurants and try really, really hard to assimilate to the dominant culture and pay their taxes; and if they can't completely be as successful as the dominant culture, they will leave well enough alone.
In contrast, Americans will look at the person sitting next to them, see them as lazy, and refuse to pay taxes to support who they think is taking all of it. That person next to them may be poor, black, old, sick, hispanic, have down syndrome, is a teacher or a cop or a union steel worker or liberal or English major or hippie... ad nauseum. Whatever the reason, that person looking over at that "other" person flat-out refuses to allow "my hard earned money" to go to whoever the hell that is sitting next to them. It just won't happen.
Another thing, our poor and elderly are heavily, and I mean absolutely, unmistakeably, and incorrigibly self-effacing, self-hating, and in flat out denial over their station in life. They are told through attack ads during elections, through cable news, and through the water cooler, that blood-thirsty communists are alive and well and have a donkey replacing their hammer/sickle emblem. They're gonna take your guns, your religion, your money, your children and your property, give it to white trash and Mexicans and Godless welfare queens who drive better cars than you on the way to the food stamp office, and you'll be left with nothing but their hospital bills that you now have to pay because they bought a flat screen and new spinning rims for their car.
No one will ever admit that the heterogeneous society that is America collectively feels this way, because we didn't have a raw sore like the Economic downturn in 2008 for us to really see how much hate we have for one another. No one cares about each other when the bill comes.