Punishment is not a harm to me. It is essential that a state maintain order for the safety of its citizens. Execution is the greatest mercy we can give some. All must understand that, as St. Augustine put it, Where there is no moral concensus, there is no law. Actions, even inactions, have consequences for the whole world.
Harm is done when an enemy is robbed of life, property, or liberty without just cause.
Plato's next weak point is that he gives the overlords permission to be untruthful. Lies are not part of a functional system. Although the power to act may be held in trust entirely at the elite level of the aristocrasy, answers do not always come from this level. An understanding of a problem does not grant a man the ability to solve that problem, but can be used to solve that problem by a man who is of the art needed to solve that problem. In short, the people of a nation are the source of the answers which the nobility alone has the power to enact.