Is there any reason to think the USA couldn't have got this by just asking nicely?
No. Not since at least Bush II, not even if Obama or Biden asked nicely.
Trump: "Hey guys, can we get some more access and stability for Greenland?"
EU: "No, not unless you contribute more to NATO, and give us credit for hosting you, and have a UN presence, and do a climate impact assessment."
Denmark: "And you pay us for every millimeter of soil your boots disturb."
Instead:
Trump: "Hey guys, sell me Greenland or I'll seize it."
Danes: "Bro WTF, stop"
EU: "WTF! We'll... defend it? Maybe?"
Trump: "You and what army?"
EU+Danes: "Ok ok, how about instead, we just give you more access and stability for Greenland?"
TES thread: "Haha Trump caved, what an idiot."
The right (and Trump) don't shit talk Europe because we hate having allies. We shit talk them because they are terrible allies. It's fine to have a weak or incapable ally; it's not fine if they pretend to be equals and make demands way outsize to any contribution they make.