The diaeresis does have an actual grammatical purpose: to break up digraphs into their separate sounds, indicating, for example, that the “oo” in “cooperate” isn’t a single vowel like the one in “loop”. Though given that English doesn’t have an “eo” digraph that produces one vowel sound, “preöccupied” is just ignorant peacockery.

The Heavy Metal Umlaut, meanwhile, stands outside of grammar and doesn’t care for it. To challenge it because “Moteurhead” sounds like a silly name for a metal band is to miss the point.