Probably woodcutter's course is eschewing "the" CA (that is, Author A's or training institution B's view of what a CA "should" look and be like) in favour of more general headings such as Vocabulary, Discourse and/or Conversation Analysis, Pragmatics, SLA, Syllabus Design etc etc - it is after all a Master's level course (right) and not aimed at (supposedly) thicko fresh graduates in need of much patronizing, simplistic spankng and abuse generally.
http://writing.berkeley.edu/TESL-EJ/ej29/toc.html (Reviews of Schmitt's, and Cook's introductions to AL)
I recall something in a link that Londo provided (or something that somebody said in one of the subsequent threads discussions) about how CA is just a load of hot air produced by academics anxious to hold onto their prestige and perks (actual power to make teachers do entirely as the academic "says", versus a general "influence" or "fame", would be debatable)...sure, some writers rarely impress or make one pause for thought, but some do, often because they are in two minds themselves about whatever issues and really do seem to be trying to see or at least acknowledge a variety of viewpoints (hence my use of "says" just then).
Anyway, like it or not, and regardless of the appropriacy always for oriental students, the main reason why CA(s) have "cornered the market" is because of the virtual monopoly in the use of the word "communicative"; and just because some schools flog hangman as "communicative" doesn't lessen the power and
import (of a proper and real consideration of) the term...it has undeniable implications for pedagogy/the starting and end points, assumptions behind and goals of, pedagogy...
Andy, I did read the whole of Sayers' essay, but can't say I was that impressed. There's her style for a start (as you yourself point out); then her reading much more into the use of the verb 'face' than the author(s) probably intended (I'm presuming they weren't being exact or literal in their usage/with the meaning - it's hard to tell given so little context):
Here is a sentence from no less academic a source than a front- page article in the Times Literary Supplement: "The Frenchman, Alfred Epinas, pointed out that certain species (e.g., ants and wasps) can only face the horrors of life and death in association." I do not know what the Frenchman actually did say; what the Englishman says he said is patently meaningless. We cannot know whether life holds any horror for the ant, nor in what sense the isolated wasp which you kill upon the window-pane can be said to "face" or not to "face" the horrors of death. The subject of the article is mass behavior in man; and the human motives have been unobtrusively transferred from the main proposition to the supporting instance. Thus the argument, in effect, assumes what it set out to prove--a fact which would become immediately apparent if it were presented in a formal syllogism. This is only a small and haphazard example of a vice which pervades whole books--particularly books written by men of science on metaphysical subjects.
Being the lazy and doubtless stupid reader that I am, I would take that at first glance (and probably second and third too) as meaning that insects are successful and endure through sheer strength of/weight in numbers (than due to any obvious advantages brought about by evolution, though they doubtless do have such advantages if we care to take to a closer look): I don't believe that insects have minds like humans, and therefore would only credit them with a "mass" mind made up of the sum of the "association"...
Perhaps more founded are my concerns over her apparent beliefs that inflecting languages are intrinsically the more valuable, and that Russian is "primitive" (that's a wooly use of a word if ever there was one - presumably she means that it is also a "valuable" or "pure" language mentally for students to learn...inflecting languages may well arguably be so, but I don't think any "normal" person ever has or really ever will enjoy memorizing conjugations or inflections for case etc by rote).
Most importantly, the essay ignores developments in politics and philosophy, which have made it hard (in at least an academic sense) for some to accept facts as the facts they purport to be (in subjects other than language anyway - and languages themselves e.g. whether to teach whatever language before or over another, and HOW, are obviously subject to controversy - as if such controversy were in itself a bad thing (it is only bad when it distracts totally from any learning taking or being able to take place, which is usually never the case)); that being said, I take no particular interest in the likes of Derrida.
About the only really reasonable thing she says is that a degree of explicit grammatical knowledge may come in handy when learning a foreign language (if only so one can understand and find one's way around reference books).
Basically, a language is not a set of facts that can be learnt purely through memorization, as the limited ability of many oriental students shows. I'm not saying there is nothing to be said for rote learning - it's certainly better than (the student, in their own time at least) doing nothing at all. Teachers, however, need to be aware not only of discrete facts (e.g. items of vocabulary, the various patterns of complementation etc) but also able to marshall those facts in a way that is true holistically to the nature of (the) language (there's metaphor to deal with for a start); there is a world of difference between knowing, as a NS student in a history class, the answer to 'Who was king at such and such a date?', and understanding the question itself (as well as knowing the answer!) as an EFL or ESL learner. (Note to woodcutter: The answer wouldn't need to be prefaced with 'The king of England at the date you mention was...'

).