I think healthcare in the US is beyond messed up. In my area which is a large city in the North East US it is very difficult to be able to get into ER because rooms are filled with people who are trying to get non-urgent care. These people have no money, no insurance and their only way to get care is to go to ER. System is rotten to the core.
I also did not get your last paragraph are you talking about the US or Denmark? I would be shocked to hear income inequality problem is worse in Denmark.