After a short time, and some casualties, I think the US military would have real problems internally, not counting that popular support would disappear.
The question is what would happen to the US staff land-locked on NATO bases within the EU. They will automatically become under siege, vastly outnumbered by European counterparts.
Since any attack on Greenland is an attack on the EU country the Kingdom of Denmark, and any attack on any EU countries automatically trigger EU Article 42.7, which mandates the full support from all members, to which all EU countries have committed, it would imply full-scale war.
Also if any french military asset is present when the US attack, we will see how determined the french military is following it's own doctrine (which dictates a 'warning shot' 24 hours before sending the tactical nukes).
My guess is as yours - the US military's focus on middle east and east Asia is of great disadvantage for them. Do they even get below -20 C for any longer periods at any base located on US mainland? Alaska, and some regions close to Canada, perhaps, leaving them with only some 10.000 personnel having anything near arctic experience, majority of which are based at the bases, not trained for front-action in artic climate.
For some real-life insides:
https://youtube.com/watch?v=Msfrit12u0M&lc=UgwDlvf-UEzzhBzZJ...
If we build a Rammstein- sized base the US would already outnumber the native population.
Would the Danes or French open fire on us while the US is setting up shop? Highly unlikely.
Trump is pushing a total takeover but I suspect he would rather leave a small pocket of southern Greenland to the Danes to continue supporting the indigenous people, and then taking the bulk of the rest for mineral rights, arctic sea lanes, and defense.
Greenland is an island full of a vast nothingness, there is enough space for those kinds of bases. Greenland and Denmark have repeatedly said as much, and allowed the US to build any number of bases of any size. Building bases is totally possible, and always was possible, because Greenland and Denmark have always allowed it and would have continued to allow that.
I mean, they even turned a blind eye towards the US loosing a nuclear reactor and contaminating quite a bit of ice while trying to build tunnels for their ICBMs. https://en.wikipedia.org/wiki/Camp_Century
This isn't about building bases or military strategy or even resources. If it were about those things then the US could take over Greenland slowly with little effort. My understanding is the population there would have welcomed investment. The US could have done some minor leg work and in 10-20 years Greenland would have been closer / keen to join, or whatever.