Bob, there are issues.
First, what gait are we talking about? The running walk is a WALK. It uses the same muscles and the same sequence as a "dog walk." The only thing that changes is the cadence (i.e., the frequency of the steps). So building muscle for walk, by definition, builds muscle for running walk (and the flat walk).
If you're dealing with a different gait (pace, stepping pace, foxtrot, etc.) the you have a different issue. The issues are related, however, and walking with these other gaits to build overall strength is a Good Thing. Specialization, for a horse coming off a long layoff, is not required.
Second, a horse that is "naturally gaited" can perform the gait even after a layoff. They will not, however, have the strength or stamina to perform that gait well or for long periods while carrying a rider. So a horse gaiting fine at liberty might well have issues when you put 200 pounds on its back.
Athletes who have been on a "layoff" often need a period of careful conditioning before they resume significant performance. Horses are engaged in athletic performance when they carry a rider. That performance will be on a scale from easy (walking around an arena) to very intense (field jumping, steeple chasing, fox hunting, etc.). Only an idiot would take a horse coming off a layoff and go ride first field. Time is necessary to build the strength and stamina necessary to the discipline in which the horse will work. This is true even if all they are going to do is slow gait down a groomed bridle trail.
Equine safety is co-equal with rider safety when coming off a layoff.