This is not about horses in general or horses in sport. This is about military power. That the king is not allowed to have many horses is to prevent him from putting his trust in military power. Horses and chariots were the tanks of that time. Pharaoh had such a force, but the Lord drowned him in the sea (Exodus 14:26–31). So this verdict to acquire many horses is to prevent the king from trusting in that power.
16 Only he must not acquire many horses for himself or cause the people to return to Egypt in order to acquire many horses, since the LORD has said to you, ‘You shall never return that way again.’