The American Saddlebred is predominantly a Trotting breed with the ability to learn
the rack and the slow gait. Many young foals will rack out in the field as it is easier to keep up with mom that way(well, my babies this year thought they were thoroughbreds and RAN everywhere they went), but once they develop their trot, they usually keep it. The true natural gaited Saddlebreds(natural in the way of how TWHs do their little gait all the time) are few and far between.
If the Saddlebred does not exhibit certain traits that are desirable in the show ring as a five gaited horse, or would look better as a three gaited or harness horse, then it is not worth the time to gait them.
There are many factors that go into deciding whether or not a Saddlebred is going to be taught. Some bloodlines favor gaited horses. The conformation of the horse, from the placement of the neck, to the back, to the shape/curve/placement of the legs. It is said that a cowhocked horse is easier to rack than a very straight legged horse. Also refinement. Usually, five gaited saddlebreds, are thicker, more massive, and can be a little coarse as compared to a trimmed 3 gaited horse who needs exceptional refinement. The shape and natural growth of the foot, especially the hind feet are great factors to consider. I have a two year old who we originally thought would make a nice gaited horse, but the simple shape of his back feet will get in his way when it comes to sliding into the gait. And then the inclination to learn. A horse that is more lateral will be easier to teach. Bloodlines also play a part in the mind, temperament and trainability of the horse.
All Saddlebreds can
learn, but only those who meet the requirements are taught. Heck, if they would make a better three gaited horse, then that is the division they need to focus on. To a lot of trainers, there is no point in gaiting a horse that will not be competitive in that division.
Here is a good article for you. Getting the first steps. How a young Saddlebred is taught to rack. | Trot.org