>It's a bit old-fashioned in English, but you sometimes will see it in older texts on words like "coöperation".

You don't even need to search for older texts! The New Yorker actually uses the diaresis in their print edition. It's certainly a bit anachronistic, but I do think English spelling can be a bit obtuse to say the least, so the orthographic hint can sometimes be useful.

https://www.newyorker.com/culture/culture-desk/the-curse-of-...

https://www.arrantpedantry.com/2020/03/24/umlauts-diaereses-...