Very interesting analyses and nuances -- each language constructs its grammatical approaches and metalanguage a bit differently I suppose, in spite of the universalistic objective (Babel!).
From where I stand it seems better not to mix (what I construe as) an 'essential' characterisation, what it is (adverb, adverbial phrase), with a functional one, what role it plays (which I would characterise broadly in French as complément circonstanciel, circumstantial complement, allowing for finer distinctions such as comitative or accompanying circumstance).
Words like "manner" or "circumstance" being used either in a looser or stricter sense add to the confusion. "Manner" is a (too?) large category as opposed exclusively to time and place, but a more smaller one when opposed to instrument or circumstance (in the narrow sense) for instance.
If anything the discussion may serve as a reminder of the analytical, reflexive, hence secondary character of descriptive grammar, which has to divide, order and classify the continuous phenomenon of usage into separate, clearly defined, but wholly artificial "boxes". Which is very often missed by apologists throwing "grammar rules" at each other, especially referring to a language they cannot read...
On another topic, I think TD's Watchtower quote is quite revealing too; the use of the verb "accompany" obviously derives from the scholarly sources the WT writer has used, making peremptory statements like "this is his own voice" (a couple of paragraphs before) stand out as non-exegetical but dogmatic.