I'm in the process of answering another, but similar, question so I can post that reply here as well when I finish doing some graphics workup.
Just to confirm, though - the E6B to which you refer is a Dalton (ie slide) nav computer, not the Jeppesen CR style ? The reason I need to ask is that there is so much variation in naming these days that it is hard to know what titles might actually refer to ...
Points I can figure from your question -
(a) if a Dalton, then you certainly will get the right answer without worrying about ETAS as ETAS has nothing to do with the Dalton vector solution. You don't do an ETAS calculation with the Dalton - it's just not required. ETAS only has any relevance for the CR solution which works on a different triangle for the solution. More interestingly, you might have a little difficulty figuring ETAS on the Dalton unless you have a square grid on the slide - certainly, I can't recall a Dalton with any trig data on the slide rule scales, such as the CR has.
(b) for the CR solution, the GS calculation is based on adding/subtracting the TW/HW component to/from the GS-resolved TAS (which we call, for want of some other more useful term, ETAS). It is ALWAYS relevant to determine ETAS for this solution. However, as the drift (wind correction angle in US parlance) gets smaller, the difference between TAS and ETAS gets smaller. As you get down to drift angles of, say, 5 degrees or less the difference is sufficiently small that we might just as well ignore it and use TAS in lieu of ETAS. Even with slightly higher drift angles the difference is not all that great.
(c) you can only use TAS for the GS calculation on the CR if the wind is aligned with the TR vector so that HDG = TR and there is nil drift. Otherwise, you must use ETAS, with the proviso that, for small drift angles, the difference between TAS and ETAS is very small and you can approximate ETAS by TAS. But, why bother ? Just use ETAS all the time and you only have one technique to apply - the answer remains essentially the same either way ....
(d) for the Dalton, there is no requirement to strain the brain by figuring wind components - the magic triangle picture does it all for you, easy peasy.