What the US needs in Greenland is access. The US already had that and any additional access could have been negotiated for without threatening war.
Countries like Denmark and the US have already agreed to not solve differences between them with force. The US breaking that by threatening war to take territory from Denmark is a huge violation that damages American soft power.
Trump claimed that Denmark has a weak claim to Greenland because all they did is send some boats there and the US did this too. This is false or misleading because the Danes colonized Greenland before the US existed. It also further violates norms and damages relations with many countries because the US is breaking agreements it has made in the past where it acknowledged Danish control over Greenland.
Other countries know that if the US uses force, or the threat of force, to steal territory from Denmark, the US may do it to them.
The US has claimed that because Greenland can’t defend itself, the US has the right to take it. Under that logic, any stronger country can conquer many weaker countries because a lot of weak countries cannot defend themselves against stronger countries, and having that territory would give the stronger country a more forward defense against other strong countries.
For a long time, the US has dealt with this problem by having weaker countries band together in some sort of treaty organization so that they can help each other defend against a strong country and not need to invade each other for the sake of defending themselves against the bigger country.
Russia and China are not a short term threat to Greenland. Getting to Greenland with a large force would be very hard and require supply chains by US, and European territory. The US didn’t seem to see Greenland as a short term concern because the US voluntarily pulled out most of its troops and equipment and kept most of them gone.
The US also already had commercial access to Greenland, and could negotiate for mining without threatening war.
Using hard power is expensive, unreliable, and often only working while you are still willing and able to use it. Soft power can create more long run influence.
The Trump administration and Republican Congressmen have been dismissive of American allies to the point of being offensively insulting and lying about these countries and their people.
Trump is just as aggressive toward good allies as bad allies. Targeting good allies who have been good and loyal, makes those countries less likely to support you in the future.
The European allies are democracies, and the European peoples do not like Trump threatening to use military force to steal territory from a European country. This will not help get pro-American leaders elected in European countries.