While I would certainly love to see google try to buy off Belgium for the entertainement value of the following ECJ case, I think you're exaggerating a bit too much

Amazon uses amzn.to when their revenue is roughly 1000x the GDP of Tonga.

My reply was mostly a tongue in cheek answer aimed at youtu.be being one of the given examples

Google already lost the right to use goo.gl for their link shortening service. I'm not sure who would be a part of a potential coup, but it seems unlikely since Greenland has no military of their own, the military being provided by Denmark(a NATO member) who has already in the past allowed the US to station nuclear weapons on Greenland, despite being a party to the NPT.

did they lose the right or did they just not want to run it anymore?

They didn't lose the rights. They still own goo.gl; go to the address and you'll get Google. They just weren't interested in running a link shortener anymore.

Yea, they probably realized there was no analytics advantage when anyone who would use that shortener would also likely have google analytics on thier site...