GTD, as presented, IS a "left-brain" tool
First, the right- vs. left-brain distinction cannot be taken too literally as referring accurately to activity that takes place on one side of the brain or the other. Some evidence suggests that, for example, new things require activity on one side of the brain, whereas habitual activities require activity on the other.
If by "left-brain" you mean 'verbal, analytical, reductive into parts, sequential, rational, time-oriented and discontinuous' as opposed to 'non-verbal, holistic, synthetic, visuo-spatial, intuitive, timeless and diffuse' (single-quoted material taken from "Maps of the Mind" by Charles Hampden-Turner), GTD seems predominantly a left-brain technique. DA explicitly leaves room for intuitive choices at a few points in the system (e.g. selecting among NAs in a context).
You could say that GTD is supposed to unburden the brain of the need for memorization of many left-brain-type items. This might "leave room" for more "right-brain" activity, but my readings suggest that there is little interference between, say, verbal memory and visual memory. DA leaves open the mechanism by which GTD helps achieve 'mind like water'. It seems that perhaps it works for some by relieving anxiety about items that need to be done later and might be forgotten. This "prospective memory" is a great weakness in our mental capabilities that needs to be addressed with mental prostheses like calendars, automated and other reminders, Post-Its, etc.