I find your analogy to be extremely strange. We are extremely well-armed. Greenland has also not been neglected in any way.

With regard to why you should care about Greenland, you've signed an agreement to do it, you are after all in NATO.

There has indeed been co-operation in protecting Greenland and this is very reasonable considering that the US has a presence there in the form of their military bases. I don't see this co-operation as begging.