I experience that most analyses of many languages that I know, in particular Swedish, (Flemish) Dutch, Norwegian and Icelandic, analyse the languages' short ⟨e⟩ as /ɛ/, while they sound a lot more like [e̞]. Is there any reason for this, other than avoiding diacritics? I feel like the distinction is required, since they are clearly different sounds to me.
I don't know of any evidence that the Swedish, Dutch, Norwegian or Icelandic languages analyze any vowels as [ɛ] or [e̞], and I don't know what it would mean for a language to analyze a vowel as /ɛ/ or /e̞/. Conceivable you could find evidence for analyzing a vowel as /ɛ/ as opposed to /e/, for example if there is also a contrast between /o/ and /ɔ/ and the front vowel behaves like /ɔ/ rather than /o/. However, a linguist may state that a particular vowel in a language is /ɛ/ or /e̞/, and then you could ask "Why A and not B?". In the case of Norwegian, there are many reasonable ways to phonetically write "e" such as e, ɛ, e̞ and even ɛə. Some people feel that one should use the IPA transcription that most-closely matches expert pronunciations, which would argue against e. Not everybody agrees with that position.
A second consideration is that IPA disprefers diacritics, which argues against e̞. The idea that the IPA letter [ɛ] has an exact pronunciation is incorrect, a letter represents a range, which is exemplified in expert tokens. In this one specific instance, you can compare 3 of 4 expert's versions of [ɛ] vs [e̞], because the lowered diacritic is in fact exemplified with e and β. But in general, there are no reference examples for diacritically-modified segments.
The best way to find out why a linguist decided on one transcription rather than another is to simply ask them. Before asking, you should think through the logic of the presumption that /e̞/ is better.