What you want is a horse that has the courage to do the job, the trainability to do what is asked and the conformation to remain sound.
Without all three you will not go far in this sport.
Thoroughbreds are fast and, if they have correct conformation for the breed, will likely remain sound. They often have a ton of courage and can move well enough to do the dressage phase. They are sensitive to train.
Remember that it is the Thoroughbred that is also the Steeplechaser..