Is Denmark selling anybody rare earths. They don't appear to have the mining, overland transport, and export infrastructure for that. Their primary industry is fishing...

My limited understanding is that they are not selling anyone rare earths. It's actually a point of contention in relations between Greenland, Denmark, and the EU's economic security goals.

For a long time, the remoteness and the lack of industrial infrastructure you identify kept anyone from being interested in mining rare earths there. That changed in the late 2010's, when China came sniffing around and started looking at possibly investing in projects there.

Bust that didn't go anywhere. Apparently, mining rare earth elements is really nasty and dirty work. It seems the mining processes end up generating a lot of environmental damage, particularly because of naturally occurring radioactive materials that are located in the same deposits as the rare earths. So Greenlanders are really skittish about it, skeptical that mining can coexist with their established fishing industry (and way of life).

Greenland is a mostly autonomous region within the Danish kingdom, so it's their call. And in 2021 their government pretty much prohibited any mining activities. Because rare earths are economically important, both Denmark and the broader EU have tried to offer regulatory incentives and protections to the Greenlanders to change their mind. With no success so far. There's apparently a desire among Greenlanders to be independent from Denmark, and the main obstacle is their economic reliance on support payments from the kingdom (their economy isn't big enough to be self-sufficient). If there were a massive windfall from mineral extraction that might allow them to gain independence, but it doesn't seem to have been enough to get them to change their minds so far.