That's pretty obvious to anyone who understands that people tend to not place value on things they get for free. Like public high school education in America.
So how does one compare the "free" (tax paid) Danish universities to the US/UK ones? By citations? Nobel laureates? Amount of foreign students? Salaries of alumni? Patents? GDP per capita? Number of high value companies per capita? I think you'd find that as a whole, Danish, and other Scandinavian universities, easily compete with Anglo Saxon universities, and that the main drawback of Scandinavian universities is the language.
When I refer to "free", I mean more than just no cost in terms of money. If there is a cost in terms of the university being very difficult to get admitted to, then humans tend to value it. This sounds from your post like it may be the case here.
American public schools are valued so little that truancy laws (for 1-8) were passed to force attendance. Large numbers of high school students simply drop out and wander off.
On the other hand, becoming a Marine is free (they'll even pay you!). But the selection process is grueling, and hence the status of getting through that is valued. Even more selective services, like the Navy Seals, have correspondingly more status and value.
In the Air Force, being a fighter pilot is the hardest to achieve, and has top dog status, despite being paid to go through the process.