I spent, oh, a third of my career as an intstitutional trainer but never as an adviser. I have worked with advisers, SF, and MTTs in operational settings. So my perspective is as an outsider with some theoretical knowledge. I would make a few comments.

1. Competent trainers can't spend their whole careers as trainers. They must be regreened or they lose competency fairly rapidly. The shelf life of an institutional trainer is probably about three years. Moreover, the 'training' world and the 'operational' world both benefit when you have a fairly comprehensive turnover. The operator who spent three years as a trainer tends to have a firmer grasp of his subject than one who never served as a trainer at all, while a trainer with no recent operational experience is less effective than one who has such experience.

2. As pointed out, not everybody can be a good trainer or a good adviser. Some can be both. I have seen 'experts' from the SF/FAO/agency world who were not effective in these roles; I have also seen 'non-experts' with no particular cultural, linguistic, or social preparation beyond what one gets at the deployment center, turn out to be exceptionally capable at bonding with their 'native' charges.

With that in mind, it seems self-defeating to try to form a seperate corps of soldiers who only 'do' instructing/advising. The 'foot soldiers' in this effort must be rotated in and out of the 'real' army. On the other hand, there should be a corps of specialists who do the screening, recruiting, organizing, quality control, and deep thinking. They could be drawn from a variety of sources into something like the proposed Advisory Corps. I think a model for this type of organization already exists in the British Army - unfortunately I can't recall the archaic name they have hung on it, but it started as the School of Musketry and most of its members seem to be Welsh LE commissions.
Anyway, this is a fairly small group whose sole function in life is to oversee various forms of intstitutional training. They are the experts not in the actual subjects being taught, but in the creation of training programs, the selection and training of instructors, the evaluation of programs, and the creation of appropriate standards. They are seeded throughout the training world and work directly for training commanders, but they are guided by a strong sense of expertise and a firm belief in the principles of their corps. They also drink like fish, but that may just be a British thing.
What we would have to add is a cultural/linguistic element and perhaps some regional orientation.