There are three good arguments against capital punishment:
(1) It undermines the general lesson we want to teach people: we should not kill each other. These sorts of lessons aren't taught mainly by rational arguments, but by making the undesired behavior 'taboo' -- unthinkable. (Like exposing children to deviant sexuality used to be.)
Unfortunately, since we fight wars periodically, where the main aim is to kill as many of the other side as possible, this is difficult to do. (Plus, as Hollywood and the gaming industry know, young males enjoy watching armed violence -- it sells. So we'll continue to be saturated with entertainment involving armed violence, mainly that where our side doesn't suffer. Unlike real life.)
(2) It gives enormous power to the state, without any possibility of correcting any mistakes the state might make. It's rather strange to see conservatives (of whom I count myself one), who don't think the state has the competence to, say, set a minimum wage or a maximum price for bread, wanting to give it the power to kill its own citizens.
And ...
'the state' is an abstraction. Imagine you have been accused of a murder you did not commit ... or of a killing that you did commit but which you, and many others, would say was justified.
Now imagine that the District Attorney, the investigating officers, the judge, and the jury ... are composed of people of a different race to yours ... people who generally don't like and/or fear people of your race. And/or that the mass media are slanting everything against you ... showing, for example, the person ou killed as a wonderful young man keen to carry on his charitable activities instead of someone with a lengthy criminal record. (See the Ahmaud Arbery case for a recent example.)
Imagine also that all these people, even if they are the same race as you, are not Hollywood stereotypes of authority figures -- idealistic, intelligent, unprejudiced, hardworking etc -- but ordinary people, with all the deficiencies that ordinary people have. Policemen keen to get closure on a case, already convinced that you did it, concealing evidence in your favor. Jury members who just don't like the look of your face, or who had a relative injured by someone similar to you -- independently of the race issue.
Would you really want these people to have the option of killing you, as opposed to imprisoning you for life, with the possibility of 'The Innocence Project' undoing an injustice later?
[
https://innocenceproject.org ] (Don't be put off by the fact that these people have jumped on the 'anti-racist' bandwagon. They also help whites who they believe to be innnocent.)
(3) It's not a deterrent. Yes, if it were to be applied immediately -- as a robber in a convenience store will apply it to you if you fail to comply with his demands -- it would be. But in real life, in a civilized country, "the law's delay" is going to take years, as it should.
And, realistically, we are not going to torture convicted killers to death on live television, even though doing that might help the deterrent effect.
And ... many, probably the majority, of killers are people with low IQs and low impulse control. They don't/can't think about consequences. Let's not explore the statistics here any further ... it's obvious to anyone who reads the news and/or looks at the FBI's Uniform Crime Statistics.
So ... how about this:
start earlier. Most killers have a history, in and out of prison, released into the community by liberal chowder-head parole boards or judges, so they can carry on wreaking mayhem. The don't usually begin by killing people ... they begin by threatening or using the threat of killing.
So ... let's do this: if someone is convicted of a crime involving use of a firearm ...
cut off his trigger finger. Yeah, yeah, 'cruel and unusual punishment' and all that, but do it under anesthetic and it wouldn't be cruel, and do it often enough and it wouldn't be unusual.