It does. The English I was taught in school, includes the diaeresis as part of the writing system. Words like coördinate, reëngage, Boötes, are spelled as I just spelled them.
I don't adhere to it. It is old-fashioned, even a bit archaic. But it is still the preferred spelling in some style guides, and it is still used in some publications.
As far as I understand, it's actually used to indicate a hiatus instead of a diphtong. E.g. naïvety is meant to indicate that aï are part of different syllables and shouldn't be pronounced in the same syllable, just like the oö in coördinate.
The catch is that the diaeresis in Türkiye does something completely different that the diaeresis you mentioned, so actually it would be less confusing if English didn't have it at all.
...then again, even some Germans are using the diaeresis for this purpose, such as this guy https://en.m.wikipedia.org/wiki/Bernhard_Ho%C3%ABcker - and you can't blame him, because usually "oe" is a transliteration of "ö", and then his name would mean "hump" (as in camel's hump).
Where did you go to school, if you don't mind me asking? I didn't realize that was ever something that was taught -- I'm curious if it's a geographic thing?
The only place I've ever seen the diaeresis used like that is in the New Yorker.
Ontario, Canada, in the '90s.
We weren't required to use it but I do recall a brief lesson on co-operate, cooperate, coöperate, etc. being valid variants.