Good transliteration is not arbitrary; it's supposed to make it possible for the reader who doesn't know the original language to reasonably approximate the pronunciation.
In your other examples, all the sounds would be considered "o-like" (rather than "e-like") in most languages with a 5-vowel phonology or similar. A reader might not be able to parse the diaresis, but if they say it with a plain "o", it's a reasonable approximation.
But then "ё" in "Пётр" is really just [ʲo] phonetically, so it should, ideally, be reflected by "o" or some derivative glyph in transliteration for readers to be able to understand and pronounce it with any semblance of accuracy. Something like "Pötr" would probably be best, actually, since in most alphabets that have "ö", it's used for [ø] or similar, which would give a more accurate approximation of [ʲo] than just [o].
OTOH the Latin "ë" (when present in the alphabet, which is much rarer to begin with) tends to stand for [ə], so it would be most logical to use it to distinguish "э" from "е". Note that this can be treated as a consistent pattern: a diaresis over the vowel inverts its "default" frontness (front vowels become back, back vowels become front) to simulate the corresponding presence or absence of [ʲ]. Thus we can then also write "ä" for "я", "ü" for "ю", and "ï" for "ы".
In your other examples, all the sounds would be considered "o-like" (rather than "e-like") in most languages with a 5-vowel phonology or similar. A reader might not be able to parse the diaresis, but if they say it with a plain "o", it's a reasonable approximation.
But then "ё" in "Пётр" is really just [ʲo] phonetically, so it should, ideally, be reflected by "o" or some derivative glyph in transliteration for readers to be able to understand and pronounce it with any semblance of accuracy. Something like "Pötr" would probably be best, actually, since in most alphabets that have "ö", it's used for [ø] or similar, which would give a more accurate approximation of [ʲo] than just [o].
OTOH the Latin "ë" (when present in the alphabet, which is much rarer to begin with) tends to stand for [ə], so it would be most logical to use it to distinguish "э" from "е". Note that this can be treated as a consistent pattern: a diaresis over the vowel inverts its "default" frontness (front vowels become back, back vowels become front) to simulate the corresponding presence or absence of [ʲ]. Thus we can then also write "ä" for "я", "ü" for "ю", and "ï" for "ы".