I think the English translation lost it's original phrase from Japanese.
"Give no rein to the spiritual horse."
sounds too vague to me, but in the sense of religion / spiritual, my guess is that it means: don't worship many ways (gods, ideas), stay true to your self (pristine origine) and follow your own path in life while accepting other's ways.
Ueshiba had said (also in video) that when he prayed, he cooperated with the many gods and spirits from all over the world, but in the end he stayed true to HIS way and to his Source. Everything has its own pace (vibration) in the living world / materials, do not hast, go with the rythm of things, and don't forget, give rein to the spiritual horse!