I don't think you can capture a >800k square mile island in the same way you can kidnap a dictator.
The question is whether the US is willing to pay the costs to do so. Sending European troops is attempting to raise the costs of invasion so that any rational actor would decide against it. Of course, we wouldn't be in this situation if all the parties involved were rational actors.
More importantly though, its an incredibly cold and unhospitable place thats inhabited by 50k people whose little kids have more arctic survival skills than US special forces and who really, really don't want us there. Unlike Afghanistan, you can't patrol the skies nonstop with drones either due to the cold.
Basically, this isn't a choice between owning Greenland or keeping all our allies. Its a choice between keeping all our allies vs getting bogged down into the ultimate guerilla war and suddenly having nothing but hostile neighbors to the north.
If Greenland really did become a state, they'd actually have a fair bit of political power. At least, until oil companies shipped in tens of thousands of employees. I can't see the Greenlanders laying IEDs for American troops, but I suppose I can see them making life very, very hard for civilians.
It is a giant island in the Arctic Ocean, populated by Inuit and people born on a pair of skis. If they don't want you here, they will make sure you freeze to death in a troop transporter stuck in the snow.
It would destroy NATO, the global financial system, our supply chains are so intertwined that pharmaceuticals, air transport, and most advanced technologies would be completely disrupted for years.
Intellectual property rights, judicial cooperation, international recognition, and all those things would be dead.
Our lives would be disrupted beyond imagination.
But it meant that the US will have fired shots at NATO soldiers, rather than just walking in and declaring themselves in charge. That raises the political and economic stakes, if not the military ones.
It's making absolutely clear that this means the end of NATO, and puts all EU/US relationships in doubt. It could even mean that we were automatically at war. They're hoping that somebody around the President will consider that too high a price to pay. Which is a long shot, but is probably the least-worst option.
Remember, the US already has treaty rights to build bases: the defence strategy of Greenland before this nonsense was "be a member of NATO, nobody would be dumb enough to attack us because if they did the USA would defend us".