Why would you say that Ä, Ö, and Ü are not vowels in German? Intuitively that doesn't seem true, but I'm not sure if linguists maybe use the terminology differently.
They're characters that represent vowel sounds, sure.
It's kinda the same thing as how in the English alphabet, Y is by definition a consonant, despite being used to represent a vowel sound in many words. (beauty, many, cyst, tryst, for example)
Of course, the more accurate take is that it's by definition neither a vowel nor a consonant because those are properties of sounds. They only apply to letters to the extent that letters consistently represent sounds, but when they do not, it doesn't make any sense to categorize letters like that.