I worded the previoust post a bit poorly because I was trying to keep it short, so to avoid creating misunderstandings, I'll post in more detail... which I probably should've done in the first place.
While え/エ by itself was /je/ and it's possible that the vowel /e/ might generally have caused palatalisation in consonants, so it could be that て/テ and で/デ were [tʲe] and [dʲe], it's not certain if there was palatalisation and certainly it was never to the same degree as before /i/ in the dialects that developed into the majority of modern Japanese dialects. The dialects that pronounce て/テ and で/デ as if they were ちぇ/チェ and ぢぇ/ヂェ developed that degree of palatalisation later. AFAIK at least some of those are also dialects that still contrast じ/ジ and ぢ/ヂ, too, so it's kinda irrelevant but I figured it's worth pointing out proactively.
In case it's not obvious, ぢぇ/ヂェ (*dye) and じぇ/ジェ (je) are pronounced identically in Standard Japanese. This kind of things are used in the transcription of dialects where ぢ/ヂ and じ/ジ remain distinct, and of course you'll come across じ, ず (ji, zu) and ぢ, づ (ji, zu; *di, *du) being differentiated in writing even though they're now pronounced identically (some dialects even merge them, pronouncing all identically with a vowel that's between /i/ and /u/... but you don't have to even think about that, I'm just mentioning it for the sake of thoroughness.
My point with ぢぇ/ヂェ (dye) was consistency with "etymological spelling", which like I said is a thing even in Japanese in the case of す (su), ず (zu), つ (tsu), づ (zu, *du), し (shi), じ (ji), ち (chi) and ぢ (ji, *di). As you've probably noticed already, in loanwords and non-Japanese names and whatnot it's less consistent and the norm is to use ジャ (ja) ジェ (je) ジ (ji) ジョ (jo) ジュ (ju) everywhere, so you'll see things like ジョン instead of ヂョン for John infinitely more, but I have seen (rarely) people use ヂ instead of ジ either everywhere or to mark an etymological distinction. Oh, and of course つ, し and ち also used to be /tu si ti/, but then that sound change inducing frication (and palatalisation before /i/) happened. Some romanisation systems still reflect that and write つ (tu), づ (du), し (si), じ (ji), ち (ti) and ぢ (di), and also ふ/フ (hu), which is nice and consistent but also kinda pointless at the same time.
By "as phonetically as possible", I meant as phonetically as Russian as possible as kind of an etymological spelling with the implication of it being pronounced distinctly from ジェ. In practice you'd be better off just writing ドブリージェン or even not marking vowel length to get ドブリジェン, which judging by Google is more common.
Also, when I said you probably won't often encounter things like ヂェ or キェ "in the wild", that doesn't apply to シェ, ジェ or チェ since those occur in a lot of loanwords from English.
One last thing, when I said "onomatopoeia", that was meant to be inclusive of ideophones and interjections and things like that as well.
>inb4 I forgot something crucial