The spoken language has more vowel sounds than the alphabet has vowels, even when Danish has added three extra (æ, ø and å).

That's one thing I think English at least did right. Other languages added extra letters and/or diacritics to try to capture all the variations on vowel sounds. But, in most languages there are far more vowel sounds than there are vowels. So, don't make things unnecessarily complicated by adding extra letters. The one language that seems to do it basically right is Spanish, where there are only about 5 vowel sounds and they use the accent character not to show that a letter is pronounced differently, but just to cue you in on which syllable is accented when it might not be the one you expect. (With a few minor annoying variations, like el and él).