Well, my understanding is that if <e> appears after a consonant, it causes that consonant to be palatalized
Only most basically. Some consonants are never palatalized (so stressed "ше", "же", "це" are just [ʂɛ], [ʐɛ], [ʦɛ]). In a great deal of loanwords "е" doesn't palatalize anything either (e.g. теннис, секс, амбре etc.). "Э" appears after consonants only in loanwords and in foreign proper names, and does so in a pretty random fashion (for example, both "тег" and "тэг" may be used in computing for "tag", and they are pronounced identically). Its most consistent role is denoting /e/ after vowels (which combinations occur only in loanwords again) and word-initially. I personally percieve the situation with "э", "е" and "ё" as the most pressing issue of Russian orthography.
and "nye" is the closest approximation from an English speaker's perspective.
Hm, I never thought about it at that angle, to be frank. Still, in Russian [Ce], [Cʲe] and [C(ʲ)je] are all contrasted to each other (and for dental plosives [Cje] and [Cʲje] are also consistently contrasted; for other consonants [j] most typically causes near-automatic palatalization).
While "ye" may not be 100% accurate, it's misleading to say that it's just "e," as though it didn't differ from <э>.
And yet after consonants it's just /e/ on the phonological level, even though it normally modifies the meaning of the preceding consonant letter. How foreign speakers percieve palatalized consonants isn't of great relevance here, I am afraid.
If there's a preceding consent it's always palatalized, to my knowledge, isn't it?
Given everything mentioned above - no, it isn't.

Even if we exclude loanwords entirely. Russian has /e/ after certain hard (non-palatalized) consonants since the 15th century, after the previously soft /š/, /ž/ and /c/ had hardened (and, of course, it's still
spelled as "e" there - no one ever tried to change that, as there was no necessity).
And again, I must remind that after consonants "э" and "е" reflect one and the same vowel phoneme /e/ (usually spelled /э/ in the traditional Cyrillic transcription) - these letters just treat the preceding consonant differently (...or not, occasionally).
The Cyrillic script was originally created for Old Church Slavonic, which had many important differences in its phonology (open syllables only, comparatively little phonemic relevance of palatalization, etc.). It suited Old Russian perfectly fine, but a millennium of phonetic shifts and phonological restructuring resulted in its usage in Modern Russian being pretty complex and sometimes counter-intuitive.