Most people are started on slower horses who are reliable and forgiving. I know it can be frustrating, but it is often for your own safety. My own trainer originally struggled to make me happy with lesson horses due to the fact that I wanted a more responsive horse and she wanted me to gain an independent seat and calm hand before moving to the more forward/sensitive horses. The less responsive horses will help strengthen your leg as well, which is crucial with the more sensitive horses. A noisy leg can send mixed signals to a sensitive horse and a calm horse will forgive you for it
I've ridden probably over a twenty different school horses in the past few years in between the fiery mare I shareboard now. Each horse has something to teach you- believe me!