You raise some really interesting points here. Not easy to answer.

The thing with killing a murderer is that modern people who never fight, never get hurt, never put themselves up to really dangerous situations and life threatening people, have a dilemma:
They lack this form of experience but of course, they prefer being it that way.

They do not hurt, push and hit others and therefore think that any form of corporal punishment is bad because they themselves have been spared from being hurt, pushed and hit. They believe they are right to refuse corporal punishment, since they don't think they can demand that others do the work for the general public. (I personally don't agree with this, as I can't extrapolate from myself to others).

Like doing the actual killing, where the death punishment exists. If I am unable to kill myself, even though the death sentence is in the law, I might want to change that law, instead of further bring death to criminals.

The thing is what one thinks of the executives and the justice system oneself lives in. Do the organs work properly? Do they judge thoroughly case by case? Do they care to work to their best conscience, ability and high standards of their education? Are they paid well? Do they earn to be honored? Is the general public appreciative of what they do? How can one find out? Do the statistics correspond what is generally perceived? Like, maybe what is generally perceived (through media sensations) might not match the statistics. And so on.

... For some, the death sentence might not be the worst to be afraid of. It might be worse to sit life long imprisoned. For others, death would be the worst. I really don't know and I have no information about our German prisons and statistics.