I agree he can never be trusted in society. Degrees of psychosis that violent are uncommon, and the person is usually beyond ever being able to fully recover. But punishment is the wrong answer. Psychiatric treatment will not only improve his state, but make him less dangerous.
I'm not disagreeing with the notion that society must be protected from violent people, regardless of the cause of their violence. I am only disagreeing on what the appropriate course of action is.
Punishment in this case is a purely symbolic, and viscerally bloodthirsty reaction. It is pointless. It helps no one. Not him and not society.
Killing people is not a rational or efficient response in any civilized nation. Societies with the death penalty have higher rates of murder. All it does is make murder socially acceptable, and thus more common. The loss of life and tax payer money over the long run is actually less by banning it, in every case.