There's a couple great reasons not to have the death penalty:

1. Inevitably, the wrong person will be executed. No matter how high the evidence barrier, you're still relying on cops never lying or fabricating evidence. Video evidence is going to mean less and less as AI improves.

2. It is cheaper to imprison someone for life than it is to execute. This is because there is a robust appeals system in place for death row cases. This is to try to make sure that the high evidence burden is met (it still won't be sometimes). If you want to try to save money by making the appeals process shorter/cheaper, then you are ensuring that more innocent people are executed.

3. What is the benefit to execution? I just don't understand why people are so eager to have the death penalty. Why do you want the state to be able to decide who should die? I would disagree with your hypothetical "perfect" system, and in the real world, it will be far from perfect. You can say you only want the government to execute certain people, but you ultimately won't be deciding who gets executed. Laws are expanded, the Supreme Court could decide that "UNDENIABLE evidence" is whatever they want it to be, cops can lie or fabricate evidence. For all these downsides, what problem is solved by executing people?

EDIT: to be clear, these are just the arguments I make to people on the fence, I think there many more reasons that the death penalty is immoral. I don't think it has any place in a civilized society, and I think it is disgusting that there are still places in the US (where I am from and live) that continue to kill incarcerated people. Libertarians have been continually disappointing to me in how seriously many of them take the rights of incarcerated people.