The satisfaction of knowing that there's one less evil, disgusting person taking oxygen away from the rest of us.
There's plenty of oxygen to go around.
Justice must act as a substitute for strong, personal emotions, not be a proxy to them. I agree with you, if someone harmed my loved ones, especially in a way or for reason that was deliberate or inexcusable, I would feel a strong desire to take their lives. Most people do. But that is why we have a system - it preserves order. If everyone settled their own vendettas on their own terms, everyone would be dead. And adding less humane means of death really does nothing but satisfy victims and families, but at what cost? Objectively, inflicting harm on a guilty person for the sake of inflicting harm achieves nothing.