You are being disingenuous. A murderous psychopath is a danger to people. This is obvious and something that a five year old child can understand. Some people cross the line and forfeit their lives. The death penalty is absolutely justifiable unless you are a moral relativist, which is a logically inconsistent world view.
I already mentioned the cult-of-personality example above and you breezed over it. Gang leaders are fully capable of executing kills from jail. They have influence that extends beyond the 4 walls they're contained to. Some of them have access to a vast array of money and dangerous people who they can delegate to. They can also actively influence or harm other prisoners in jail.
Treating true psychopaths with kid gloves won't make the world a better place.
If prisoners are able to lead criminal organizations whillst inside, that's very obviously a rather trivial-to-fix bug in the design of the jail, not a reason to murder someone for vengeance.
We have more than adequate technology to keep such hopeless cases from harming themselves or others.
I believe that this is a false dichotomy that you present; I can think of several different solutions to this problem in seconds, none of which are what you describe.
The fact that you move on so deftly to the word "deserve" suggests that it's vengeance you seek.
An eye for an eye makes the whole world blind.