Sure - they have to follow what they've already agreed to.

And are they? The answer is NO in many cases.

But in their internal disputes with Denmark and the EU, why wouldn't they pursue "Greenland First" in any matters that they're not already obligated to do so, to the same extent that we are changing our foreign policy approach to being "America First"?

It's in America's interest to keep the Atlantic sea lines of communication (SLOC) open.
It's in Denmark's/Greenland's interest to keep the Atlantic SLOC open.

It's in America's interest to not have Russian and Chinese Naval and air activity operating with impunity in the Arctic.
It's in Denmark's/Greenland's interest to not have Russian and Chinese Naval and air activity operating with impunity in the Arctic.

It's in America's interest to source and trade certain strategic materials from secure, trustworthy partners.
It's in Denmark's/Greenland's interest to source and trade certain strategic materials from secure, trustworthy partners.

Hmm. It seems that "America First" just so happens to coincide with a lot of "Denmark first" or "UK first" or "Germany first".

But if they reverse course and decide that they'd allow mining by whatever nation offers them the best deal ("Greenland First!"), why wouldn't they follow their national self-interest rather than prioritizing America's interests?

Sure. They could sign on to the Belt and Road initiative full-bore. They could take all kinds of Chinese money, let the Chinese start transporting in loads of stuff into Greenland.

Maybe the Chinese start dragging their anchors across comm lines the way they are in Norway and in Taiwan, cutting vital communication links while they're at it.