It exists to indicate how a word is pronounced. Naïve is a better example IMO, cooperation feels too familiar.

Non-native speakers might see something like "nave" instead of "nigh-eve" unless it is clear that there is a stress that breaks out of the diphthong.

I don't think style guides are (usually) about absolute correctness, but relative correctness. A question is asked, a decision needs making, someone makes it, and now a team of individuals can speak with a consistent voice because there's a guideline to minimize variation.

IIRC it's use is to distinguish vowels that belong to separate syllables with vowels which form a diphthong. I think this could be beneficial to language learners, to give them a hint that cooperate is pronounced "ko ah puh rayt" instead of "ku puh rayt", and likewise naïve as "nah eev" than "nayv" or "nighv".

You’re replying to a troll - their entire argument was circular and self contradictory.