I’ve not read your whole post but I’ll give you the short answer. It is the duty of government to protect righteousness among people. To keep society clean. Such is commanded by God.
Shedding innocent blood...ie murder, is NOT commanded by God.
One punishes evil and severe wrong doing. One IS evil and severe wrong doing.
Execution is a gift to the person knowing he will soon meet his maker. If they live on and on they likely will never repent and come to the knowledge of The Lord and Salvation. A time limit gives them time to think, or not.
I can’t remember where it is in the Bible, but Father says “You have these evil things in your midst because you do not put them away from you” paraphrased for sure. Like tolerating and keeping molesters and rapists alive, put them down and they will cease to be a problem.