This isn't a potential war against some unheard of country on the other side of the world with a communist regime. This is a potential war against friends over a territory a majority of Americans doesn't support invading.

It doesn't matter how much shock and awe the US have when the population will lose their shit and topple the government. I bet the US could occupy Greenland within half a day but a couple of months later the new US administration will beg the EU to tell them exactly how much they need to grovel for this to be forgiven.

Sending EU military to Greenland was not to intimidate US - how could we? It was about delegitimizing Trumps claim that Greenland could be claimed easily by Russia or China.

Initially deployed were a small number of troops, about thirty personnel.[10][11][12] However, the initial military planners, reconnaissance teams, were not intended to deter US action by themselves alone. They were to coordinate the arrival of larger land, sea, and air forces publicly pledged by France, and/or other European states: "As well as providing a show of political support, the European troops were said to be on a short scoping mission, according to one country involved. The aim was to establish what a more sustained ground deployment in Greenland could look like, partly to reassure the US that European NATO members were serious about Arctic security".[13] The troops also act as a tripwire force, such that an attack against them would be considered an act of aggression against the European partners and reacted to accordingly.[14]

https://en.wikipedia.org/wiki/Operation_Arctic_Endurance

But they do fuck up a lot.