I'm not for death penalty. I believe this isn't a way to do justice in a way that serve a purpose. But I'm not fondamentally against it, as if it was some moral bastion that couldn't be touched on. As for giving the judicial system such a powerful, the judicial system already has powerful tools.
You can't overturn a wrong execution, but you can't overturn a wrong imprisonment after 40 years. Even if you end up recovering your freedom... no one's going to give you those 40 years back. Your life is pretty much fucked up if all you've been doing since 20 to 60 was to be in a jail.
Then again, one could argue that death penalty could only be used in cases where there's not a litteral doubt that the murderer was caught killing. But with such speculations, we could rewrite a perfect world. But the reason I'm arguing on that topic is the following: While people question the right for the justice to kill or not, they never do so when it comes to military.
When you think of it, when a police officer or a military opens fire on a person, they become both a judge and a party and they decide by themselves to kill the person in question. In fact, it all comes down to "we should be better" line of thinking. Death penalty should be abolished worldwide because it's a useless and outdated system. It brings no justice, if anything it brings an escape to someone who will never have to face the consequences of their acts anymore. It also cant be outdone as you said, in case of wrong convictions.
But the line of thinking that "it's barbaric" "it's not moral" "we should be better" sounds really off to me, almost dangerous to where that kind of thinking is going. After all, detention is immoral. Restraining one person's freedom is immoral. Because they may be a danger to the society ? Well, one could say we cant foresee upcoming events.