But executions change all that. Whether or not a person supports capital punishment reveals if their morality is a flimsy thing of feelings and self-interest or a thing of principle. It is safe for someone to hate a heinous criminal and wish them ill. There's little fear of guilty feelings or reprisal. The average person will give in, baying for blood purely because they want to see the hated person suffer. A good person will refuse to hurt others no matter how badly they want to.
Capital punishment is a good litmus test. It reveals those who are murderers deep down inside.
Does this include or exclude killing in self defense? Would a good person allow themselves to be killed rather than harming the attacker? Imagine that the assailant would be rehabilitated after serving 10-15 years for your murder and live a productive life with family and kids, does that change the equation? Is absolutist pacifism the end goal or is there an arbitrary line you are willing to draw?
Executions are different. People are not afraid to admit that they support executions because they like knowing that hated people will suffer. That is not something I will consider acceptable under any circumstances. If there were any evidence that capital punishment served a purpose beyond feeling good I might change my tune. But there isn't, so I won't.
You yourself admitted executions serve no practical purpose, so your self-defense hypotheticals are irrelevant.
We should not torture them on purpose but besides that I do not particularly care if they suffer or not during execution.
As bad as that may be, it isn't a sign of violent intentions.
There are a lot of people who aren't shy about supporting the death penalty simply because they want suffering and death. There are others who don't care what's happening in the world, give in to peer pressure, or honestly believe executions have practical value. Those aren't the same problem as simply cruelty.