Truthfully we’ve never seen the US engage in a total war to conquer a nation.
Ahem: Vietnam.
Also I think you're missing a massive point here: You can't "topple" Greenland by totally destroying or taking over Greenland because it's part of Denmark and the seat of the Danish Government isn't there, nor are their main military assets, and this is before you even consider their European allies.
If Greenland was a country relying only on itself, it would totally make sense that it could be taken by just taking its major cities, but it's not, it's an independent region of Denmark, a country which in turn is allied with almost all other European countries.
The US can invade and totally crush Greenland's big cities and that will still do very little to crush resistance because that's not were most of it will be coming from. This also brings us around the whole carrier group thing: the carrier group would be how the US would be trying to stop the feeding of resistance in Greenland from Europe, since that would be coming from the very opposite side of the island (and as "islands" go, Greenland is huge, with 25% of the area of the US, so that's a pretty insane task).
IMHO what the Greenlanders and more in general the Danish should be doing is not to try and stop the elements of warfare that the US does best - such as the actual initial invasion - but actually try and make that as costly as possible whilst at the same time setting up the conditions for a long term Resistance effort from the areas outside the cities to turn Greenland into a graveyard for American soldiers, something which is far more likely to end up with an outcome like Vietnam were the daily procession or american coffins turns an overwhelming majority of the population against the War and the end result was that America ultimatelly lost it.
Finally on the last point, fighting Greenland is fighting Denmark and there are way more people in the rest of Denmark than in Greenland. That said you are right that many (if not most) of the people living in Greenland who know the whole place including the hardest and most remote areas, are probably descendants of the Innuit rather than of the Vikings (both people colonized the place).