"I don't understand. He dies at your hand before he does physical harm. "
Again, I have no problem with killing in defense of self or others in an immediate situation. A knife at someone's throat is an immediate situation. I would shoot such a person at once. The act is not necessary for self defense, only the intention.
But, the situation must be immediate and plausible. Someone holding a gun to someone's head, a knife at their throat or actively attacking another person with intent to kill presents an immediate situation which justifies defensive action.
Preserving my own life and that of others is my only concern in an immediate situation. In all other situations, I may have other choices. An execution has nothing to do with immediacy. Therein likes my objection to the practice.
Fair enough. But you would personally "execute" (in self defense, but execute nonetheless) someone before he committed murder. How do you know for sure he was going to murder your child? He may have had no real intention to do it, so you would have killed a man who was only there to rob you. You would not allow the state to do it after he committed murder. It is infinitely worse after he committed the act. Why do you think you deserve the power of God but the state doesn't deserve it?