Greenland's per capita GDP is 13th in the world. They're doing fine. Greenland has also always had a small population. It's hardly fair to call it dwindling, when it's about double today what it was in the 60s. Also a large part of the emigration is because the choices for higher education are very limited. Not because of a lack of industry or employment. As there is more than enough mining, fishing and tourism to sustain a population of 57000.

As for shipping the ice, I'm sure you aren't going to try to tell me that keeping ice refrigerated from nuuk to Dubai as well as the extra fuel used to transport (ships displace more water when they have cargo) have less of a carbon footprint than simply freezing water in Dubai.

Some Greenlandic music for you