I agree that the best condition for a running walk is miles and miles of walking, and that trotting will help fix a pacey gait. But, if trotting is all that a trainer is asking for, it's not going to refine the other gaits - the easy gait (whatever that is for the horse), the solid walk, and the canter. It's also true that the more gaits your gaited horse can do, the better they will handle long trails and remain sound for longer. If they can have 5 or 6 gears instead of 1 or 2, it's great for their back and abdominal muscles and joints to get the different workouts that the different gaits provide.
From another perspective, I can say that being comfortable with trotting is an asset for other things - uneven ground, over poles, etc. I took my TWH mare to a cowhorse clinic this weekend and it would have been great to have her do more of her trot in the pen with the cattle!