Umlaut modifies how a letter is pronounced; diaeresis tells you where a new syllable starts. I'm not sure how to pronounce "u" or "ü" in Turkish.

The problem with naive is that it's not very clear how to pronounce it: it's "na-ive", which isn't very clear. With naïve there's a clear indication that the "i" starts a new syllable.

While I agree that diaeresis in English are more archaic than not, I don't agree it's better style to leave them out. Right now there is, in my opinion, no good way to spell some words: "cooperation" is ambiguous (coo-peration, coop-eration, or co-operation?), "co-operation" looks ugly and is an abuse of the hyphen, and "coöperation" is nice but rather archaic.

A bit of effort to reduce ambiguity and match spelling and pronunciation (as far as the language allows) doesn't strike me as a bad thing.