I had someone tell me that you can tell by the pedigree of an American Saddlebred whether they are gaited or not. I didn't know this, although recognized that some saddlebreds gait, where others do not.
I would say from looking at bloodlines you can tell possibly what you may see later in a foal in their show career. I would have to guess that the horse you have is either pleasure, park or three gaited from it's lines. Both sides have produced some nice gaited horses as well as three gaited horses. In the end, it's what the horse can actually do in training, not what it was potentially bred to do. I have never seen a Saddlebred amble on their own either, they have to be taught how to slow gait & rack even though it is in their breeding.