What do you think of this analysis from reddit?
The Judge is the avatar of a paleoconservative entity called Death which would see all consciousness expunged, after being rendered completely irrelevant. All things reduced to a self-swallowing process, causes and effects dancing dumbly around the void, an ocean of blood and froth that circles a black hole like a drainspout, on and on until nothing is left but the entity’s reality — none other now possible —and its name, the last word, phonetically sounded in the last language that remains. SILENCE.
Being an avatar, an embodiment, the Judge is able to think and act strategically.
It’s probable he wants the Idiot to have some effect on other men. He intends to use him as a tool, a totem of animal regression. Therefore the leash. The Idiot is of course a person too, but that personhood is what the Judge wants to render utterly useless. Think from his perspective if you can: he can let the Idiot die, or pluck him from that stream of events. If the Idiot died in the river, men might sympathize with the Idiot, who was fascinated by his image in the water, an experience to which they can relate. That is not acceptable to the Judge, so he saves him. I am sure he can be destroyed later in a way that, this time, signifies nothing.