If the electrons are travelling at saturation velocity ,whatever the voltage is applied then there will be a fixed number of charges coming out of the material at a given time.Fixed number of electrons means fixed charge per unit time, which means constant current independent of voltage which is not the case .
Thus i think saturation velocity is just an upper limit ,when voltage applied is exceeded the electrons can not exceed the saturation velocity.For normal cases electrons will moving at speed which is dependent on voltage applied