Why would Putin enjoy a heavy US naval presence in the contested Artic Circle waters?

In theory, he might not like a heavy foreign naval presence, but Greenland is already NATO aligned. So he’d be trading a NATO aligned region for a US-only region in exchange for a fractured NATO. Sounds like a decent trade. Also keep in mind practically speaking they are probably equally unlikely to actually boost military presence much, and if they really wanted just that, NATO would probably mostly let them do it if USA paid for it, without USA having to take it.

The Danish are not meaningfully contributing to the Ukraine conflict. And there is no reason to believe the big EU militaries would stop feeding supplies to Ukraine if the US invaded Greenland.

This isn’t about just Denmark, which materially would barely be impacted right now since Greenland is doing practically nothing. It’s about the notion of one NATO member invading another, and the absolute clusterfuck that would bring. A USA versus European NATO scenario would be his dream scenario. See “Foundations of Geopolitics”, where it’s mostly about getting Russia’s opponents to fracture and ruin alliances:

Russia should “introduce geopolitical disorder into internal American activity, encouraging all kinds of separatism and ethnic, social, and racial conflicts, actively supporting all dissident movements – extremist, racist, and sectarian groups, thus destabilizing internal political processes in the U.S. It would also make sense simultaneously to support isolationist tendencies in American politics”.

This is also the book that makes annexation of Ukraine the top priority, that it must be secured before the broader Russian agenda can be executed. It also says they need to make the UK isolated from the broader EU. They know that above all else alliances must be broken for them to stand a chance to seize power.