Are you absolutely positive? Danes and Germans already sent troops and France a nuclear sub and a Frigate. UK, Sweden, Finland and others are preparing to send troops as well. Perhaps "small operation" is not a foregone conclusion.
80% of the place is frozen ice - you would need specialist units to fight there - US has never done that really - the Russians had units equipped for arctic conditions and they were locals and had unique armor meant for that (tracked-lighter-more durable).
Even China has more experience due to their conflicts in the Himalayas and Tibet.
You not going to get a Stryker or Abrahams tank working there.
Good point. Ukraine example:
"Battlefield Challenges (Abrahams): Despite being formidable, these tanks struggled against Russia's extensive drone warfare, leading to high attrition rates, with nearly 90% of the original U.S. fleet lost or damaged..."
Whatever military capability there is in NATO, it's clearly on the side of the US. The EU can't even produce basic things like artillery shells and explosives at this point. The UK can't even make steel.
I don't think America ever said that Greenland would be a state, that it would be an overseas territory of the US instead, which still sucks, but I don't think Trump wants 2 senators and 1 house member from a territory that is bound to be much more liberal than the average American state. Inuits vote Democrat even in Alaska.