An example from OLD:
As always, he said little.
On the website, as is classified as an adverb. But the full form of the first clause is
As he always did,
I wonder whether in this full form as is still an adverb.
My intuition tells me as is a conjunction that introduces the clause.
The fact that "as always" amounts to the clause "as I/he/she…people/etc always do/does/did" shows that the meaning of the combination of "as" and "always" cannot be obtained from the separate meanings of the two words. This is the clue that allows one to talk of this phrase as an idiomatic expression. There is a confirmation of this fact here (Two Minute English).
Etymology and Definition of “As Always”
The phrase “as always” combines the adverb ‘as’, which describes the manner something happens, with the adverb of frequency ‘always’, meaning every time or always without exception. It serves to communicate that a situation or behavior is recurring just as it has in the past, indicating an anticipation of consistency upon each occurrence.As a phrase, “As always” conveys that a situation or behavior is recurring as it has always occurred in the past, a sentiment anticipated upon each occurrence.
An adverb of frequency, like “always,” is an integral part of this idiomatic phrase. Adverbs of frequency express how often an action or situation occurs and can include words such as “never,” “rarely,” “sometimes,” “often,” “usually,” and “always.” Coupled with “as,” the phrase “as always” gains a nuanced meaning and emphasizes the consistent, predictable nature of something happening without fail.
It should be remarked upon that other adverbs of frequency do not necessarily combine with "as" so as to yield a similar idiomatic expression.
"As sometimes" is not used; instead, you might say "as sometimes is the case".
"As ever" is an extent expression, and so are "as usual" and as often.
There is the choice of "as often is the case" in place of "as often", but no such possibility exists for "as ever" and "as usual".
Addition due to remarks by user Andrew Leach
"As often is the case" is found in plenty (more than a hundred hits in Google Books); however, much more frequently, what you find is the same expression (same meaning) with a different word order: "as is often the case". The following ngram bears witness to that.
The same thing is true for "as is sometimes the case" (ngram below).
Also, "as is ever the case" is comparatively rare but a perfectly acceptable option.
