Good find - yes there is private health insurance in Denmark, but its primary role is to get ahead of the waiting list for non-lethal operations such as bad knees, new hips, etc. And you get cushier treatment since you pay for it of course. But with the government program you can always see a doctor (I was there the other day, and I got an appointment the day after I called) and you will always get treated for whatever illness you have. The private health insurance will pay for glasses, etc. and it costs around $40 a month.
Yes, that's what Canada has as well. We've got an EHC benefit that is pretty standard that pays for drugs, private hospital beds, etc etc.
Nobody is left out in the cold, but to properly compare it to the US's medical insurance, you need to include the private portion as well to equate equal service.