My two cents, based upon my own (admittedly limited) experience as a novice instructor, is that many people, both instructor and non-trained gun toter (and I'm not referring to anyone here, rather the gun toting outside world) tend to pick a single position, be it T.I., Sul, Low Ready, etc, and view it as a one size fits all solution, not considering that situations are fluid, not static. I've never been through a class with Costa, Haley, Vickers, etc, so I have no clue what they teach, but from what I've seen from most of the local yokels around here, none of them convey to the student that different situations can call for different methods of handling the gun.
I've been through two LE academies (federal level and local level), and neither taught moving with the gun out. They taught low-ready for standing with the gun out, but not pointed at a target, and that was it. For running, it was "gun in holster, retention engaged, hand on gun" in both. In-service "training" at my current agency retains that same dogma.