'Coördinate' and the likes make perfect sense to anyone who started out with a language such as Dutch or French where this is common. It's rather refreshing to see it used in English, where usually even the humble 'resumé' must beg for its accent aigu.

It never confuses me though, despite being able to read German as well. The mind just shifts to whichever language you're reading and interprets any diacritics accordingly. English just doesn't seem to fall under the 'those are umlauts' category of languages for me.

It certainly helps that there are very few words in English that use umlauts, and even fewer where an umlaut following a vowel could happen. I guess theoretically you could form the word Astroübermensch (who knows, maybe that's what Musk will call it when he selects the best people to send to Mars). Then it would be ambiguous whether you mean the übermensch spelling and pronunciation or the more common ubermensch. But that's a stretch

In Dutch I appreciate the diaresis in zeeën and the like. Can’t think of an English example though.

Naïf... if you can call that English.

It's English. Naïve is more commonly known.

Very few people would write the dots, but I wonder if that's more because they don't know how to type it than anything else.

Here's a list: https://github.com/RichardLitt/diaeresis-word-list

Maybe "cooëeïng", though I don't think I've seen that written with diaereses (probably not a very common word, but it's a nice example of a long vowel letter sequence, and one that people from NZ/Australia should understand).