(Pro 26:3 KJV) A whip for the horse, a bridle for the ass, and a rod for the fool's back.
I don’t think there is any significance in the three tools of control. It is not a hierarchy. That is not the point. I am not a horse person but I have read that horses can be trained in such a way that you don’t need a whip. That does not mean that some don’t. I have never heard of a donkey that would cooperate. Fools are all around us and don’t seem to respond to the subtle forms of leadership. The point is that many people in our population need force to control them.
In our modern thinking about equality, this does not sit well. We do not want to admit that “fools” (3864) exist. We do not want to admit that some people should be treated different simply because of who they are. I think the message is that fools need someone to control them and it can’t be done with finesse.
There is also a repeated Biblical standard that says physical punishment is acceptable. This does not mean abuse or torture, but it does mean inflicting unpleasant force. Perhaps if we had not abandoned this idea we would not see so much violence in our world.