If Denmark could come "steal" something like the Pine Ridge Reservation in the Dakotas, it would be an imperfect analogy. As long as we could lose it while still saving face, it would be absolutely amazing economically for the USA.
IMO it is foolish both for USA and Denmark to take them. Cut them loose as they desire and let them trade their minerals which they will surely do as soon as they realize their free lunch is gone. You would still get the thing you want it for, the greenlander are happy they are free of the colonizers, and it likely wouldn't change things militarily as they would absolutely sign a defense treaty as soon as you hand them the pen.
This is basically a fight between government of Greenland, government of Denmark, and government of USA all acting generally against the wishes of their own populace. And somehow the dumbest of all options, the USA acquiring them, seems to be one of the more likely scenarios that actually ends up happening.
Not only has Greenland never been part of the USA, it has been part of Denmark for longer than the USA has even existed.
The protection of the population and the illegitimacy of the current government was an argument developed by Russia at the time, it has not been yet by USA but I suspect this might start develop in the next few weeks.
The common ingredients to justify an invasion/annexation is a mix of: - Self-Defense, security - Historical, Geographical claims - Protection of the population - Moral sugar coating (we had no choice)
Honestly, it is such a surprise that the difference is not obvious.