But my point is that are you just "ignoring" the other side of the argument - if you don't put a serial killer to death, there's a chance others will be killed.

But my point is that are you just "ignoring" the other side of the argument - if you don't put a serial killer to death, there's a chance others will be killed. If you don't put them to death, can you be 100% sure saving that life isn't going to cause more people to die? It's not simply a case of "punishment". It's preventing further crime and deaths.