Phonetics of ä vs ö and ü
Like other vowel letters, ä in German represents two sounds, a short vowel and a long vowel.
The short ä sound is not a "modification of" the short e sound. They make exactly the same sound in standard German: [ɛ] (so Männer and Ende or bellt and fällt both have the same vowel in the first syllable). The short vowel sound [ɛ] is written differently in different words as a spelling convention (the difference is partly based on etymology, and partly based on other things like the existence of perceived relationships between words).
The long ä sound is harder to talk about. It is prescribed to be something like [ɛː] in Standard German, a long vowel different from both long a [aː] and long e [eː]. It has been argued that the distinction between [ɛː] and [eː] in Standard German is in some way artificial/unnatural; I don't know that much about the actual situation, though.
The difference between [eː] and [ɛː] is not particularly analogous to the difference between o and ö or u and ü. The standard linguistic description is that o and ö or u and ü differ in how far forward the tongue is when pronouncing them: The letters ö and ü represent what are called "front rounded vowels", and the letters o and u represent what are called "back rounded vowels". The difference between [eː] and [ɛː], both front unrounded vowels, is instead a matter of how "close" the tongue is to the roof of the mouth: [eː] is a "closer" vowel and [ɛː] is an "opener" vowel.
German spelling isn't based on the phonetics of the modern language
In any case, even if your phonetic analogy were correct, it wouldn't have that much relevance to the spelling of German, because many parts of the spelling of German are not based on the phonetics of the modern language. For example, the use of the digraph ie for the long i sound and the use of doubled consonant letters to represent the shortness of a preceding vowel sound come from pronunciation features of older stages of German that have been lost in Standard German.
The use of ä rather than ë follows the logic of etymology and function: ä is etymologically equivalent to ae (the double dot diacritic developed as a form of the letter e placed over the preceding vowel letter), and therefore is used in place of ae in certain words from Latin like Präsident. And in native German words, ä functions as the replacement for a in words that display the morphological process of umlaut: e.g. the form Männer functions as the plural of Mann and the form fällt functions as the third-person present form of fallen.