I think Trump just has a fairly transactional world view, including to international relations.
He recognises the strategic importance of Greenland, but if American taxpayers are going to pay for the defence, then he expects to get access to the resources- to make it worth their while.
It's actually pretty similar to how the European powers related to their colonies: they invested money and expertise, but certainly expected to make enough money out of it to make it worthwhile.
Weirdly, in the UK we've moved to a worldview where we expect to still invest and finance other countries... but without getting anything material back from them. A patron 'big brother' type relationship. And in our turn, we expect the US to finance our defence, be our patron.
It's a remnant from post-war thinking, which has lasted longer than anyone could expect but isn't really sustainable. Countries - their politics, their development level, their trade, their strengths - have changed so much in the last 80 years. Geopolitics is bound to change too. We would be wise make political decisions based on reality rather than based on a fantasy from the past.
That doesn't mean rolling over for the US. We're not powerless, as a bloc. But it may mean acting in a more transactional way ourselves. In particular, not getting into debt and giving away our childrens' futures for the benefit of other countries: with nothing expected in return.