I don't think that that quote was about the adverbial parenthesis only. My understanding is that all parentheticals are functionally adverbial, whether or not some form of punctuation sets them off from the rest of the sentence.
Here, simply for context and not for grammatical rigor, is how my digital Merriam-Webster's 11th Collegiate Dictionary formally defines an adverb and--by extension--an adverbial:
adverb
: a word belonging to one of the major form classes in any of numerous languages, typically serving as a modifier of a verb, an adjective, another adverb, a preposition, a phrase, a clause, or a sentence, expressing some relation of manner or quality, place, time, degree, number, cause, opposition, affirmation, or denial, and in English also serving to connect and to express comment on clause content — compare ADJUNCT, CONJUNCT, DISJUNCT
As we can see, an adverbial is a catch-all modifier for an entire sentence, regardless of the specific part of speech--by definition it could be anything whatsoever--that the modifier specifically modifies in that sentence. This being the case, every parenthetical works as an adverbial in a sentence, and in this sense we can actually consider the term "adverbial parenthesis" as a redundancy. With due respect to H. W. Fowler, to use "parenthesis" alone would be enough--but of course he was a prescriptivist at heart and might have been worried that to do so might incline learners to think that "parenthesis" referred to the pair of curved marks alone.
As to the choice of the double-dash as punctuation for certain types of parenthesis, it's true that it could look like "a tad over the top" to some people; indeed, some newspapers find the em rule more convenient to use. But I think that the choice is a matter of style rather than a matter of grammatical correctness or wrongness. I prefer the double-dash simply because it seems to me to truly link the parenthetical physically to the rest of the sentence in the precise sense that's intended for a parenthetical. In contrast, the em rule requires the parenthetical to be separated from the rest of the sentence by an em space up front and at its tail end--a grammatical contrivance that I find somewhat clumsy and rather cumbersome to execute with the keyboard of my laptop.
And, oh yes, you're right, maxsims, the "beat up Strunk" bandwagon is indeed "taking on more and more passengers." I'm not exactly a passionate passenger of that bandwagon but I do believe that with age should come grammatical wisdom, whether the subject is English grammar or the fact that the Earth revolves around the sun rather than the other way around (for which belief many religious zealots had burned at stake quite a few scientifically minded thinkers).
As to the "simple but egregious error of subject-verb agreement that remained in the [Strunk and White's] book for some forty years," it was probably because there weren't many eagle-eyed maxsimses yet at the time. Now we have at least one in this Forum, and I sincerely believe that English grammar and the language in general are the better for it.
p
