Were they al demonyak?†

In "The lady was al demonyak: historical aspects of Adverb all", Buchstaller and Traugott (B&T) consider the "development of the functions and structural properties of Adverb all, and suggest that degree modifiers in general should be analyzed in similar terms" (p. 345). They claim that the authors of CGEL mention "Adverb all as a ‘modifier in an AdvP’ that means ‘completely’, but do not subclassify it in their discussion of types of adverbs" (p. 345). The reason that all is not subclassified as a type of adverb is that it's not an adverb at all. It's a determinative.

This is clearly stated on p. 549, where it says, "the determinatives that function as degree modifiers are illustrated in [37]," and then gives [37x] He seemed [all confused].

Note that the x in [37x] is the Roman numeral, which is to say that all is one of 10 determinatives—it's the last example—to function as a degree modifier. The category of determinatives is small, comprising 35 or so members (depending on how you count and excluding the cardinal numerals). In other words, a sizable minority of determinatives have this function. Now, you could posit adverb counterparts for each of them, as B&T do, or you could just say that this is one of a number of functions that determinatives can take on.

Of course, if you regularly fail to distinguish between category and function, you might assume that determinatives function only as determiners and overlook the generalization that a clear category-function distinction allows you to capture.

At any rate, either B&T didn't read the parts of CGEL that they cite, or perhaps they didn't understand it because they had some preconceived notion of what could function as a modifier. Or maybe the devil made them do it.

References

Buchstaller, I., & Traugott, E. C. (2006). The lady was al demonyak: historical aspects of Adverb all. English Language and Linguistics, 10(2), 345–370. doi:10.1017/S136067430600195X

†B&T gloss demonyak as "possessed by the devil" so al demonyak would translate so something like "all possessed" or "completely possessed".