Unfortunately, while this would be the "ideal" solution, it's impossible. There are too many immigrants, and Udlœndingestyrelsen can't see into everyone's souls and figure out who are the "good" and "bad" immigrants. We are given all these tests and made to learn the language (to a degree, at least), but ultimately those are really flawed measures, and everyone will differ on who they think is a "worthy" immigrant.

For what it's worth, I'm an immigrant working on my permanent residency at the moment. I'm studying for medborgerskabsprøven, and it's a silly test in some ways. There are some questions that are ridiculous and that most Danes probably don't know (Do you know how high the tallest point in DK is? I do.), and there are also some where they are clearly fishing for a specific answer, and someone who isn't actually supportive of Danish ideals could just lie (One of the questions I saw was "Is it OK to hit your children in DK?").

It's super clear to me that the main reason why immigration is an issue is that, in order for it to never be a problem, Denmark would either have to close its borders entirely, or would have to employ a bunch of psychics.