The only argument I can think of counter to the idea of taking Greenland through, say, hostile trade policy and sanctions would be that Greenland simply isn't ours to take.
But big real life facts:
Greenland harbors enormous natural resources that are mostly unexploited and unknown and ready for prospecting
Greenland is probably almost indefensible by Denmark, a tiny nation with a small GDP.
Being indefensible: if valuable resources were discovered in Greenland, then being a territory of a tiny nation, even with NATO support, it could be readily invaded or exploited by any powerful enough country. China, India or Russia, easy.
Being indefensible and with murky legal status, no commercial interest would want to risk long term large scale exploitation of Greenland natural resources without guarantees of security for assets and people, since a discovery, say of lithium could cause other countries to perk up interest.
Antarctica was in a similar legal status and it was carved up into pie sliced wedges for many of the world's powers as special interest zones/sorta territories. You could argue that maybe Greenland should be handled the same way but OTOH the world is not what it was when the treaty was signed in 1959.
If you read the map legend on this map:
https://upload.wikimedia.org/wikipedia/commons/2/26/Antarctic_political_map.png
It says that neither RU or the USA recognize other country's territorial claims. Russia and the US are the shitheads of the world.
Since Greenland is not likely to be exploited for natural resources without direct stewardship by a powerful enough country, then IMO the US has a moral pretext for a takeover of sorts. It's either we do so or Russia or China (more likely China) would overrun the coast and defend it as new territory. It may as well be us.
In return the US would provide security guarantees to commercial operations and I imagine would provide very generous security and foreign aid assistance to Denmark as compensation.
As it is now Greenland is a wasteland, unused.
This of course assumes that "higher use" (as lawyers and realtors like to say) is a priority over just letting the pretty glaciers that nobody can travel to anyway stay pristine.