So the incompetence of prisons is what makes you in favor of executing people?

https://www.nbcnews.com/news/us-news/colorado-crime-analyst-manipulated-dna-data-sentenced-rcna596722

How do you prove the video hasn't been tampered with?

Where is the line between "objective and irrefutable" and "room for reasonable doubt"?

The words you're using have rules. I don't mean this in an adversarial way, but you are wrong.
Irrefutable evidence does not prove the claim, it just proves that you can't refute the evidence. Video evidence can be misleading or fake. DNA evidence needs context to prove anything.
Anyway, like I say, these things have rules. The only things in the entire human domain that can be positively proved are mathematical statements. This is why gravity remains a theory, even after 350 years. One single counterexample disproves it. There are numerous examples of this happening in the context of the death penalty, where a jury was so convinced of guilt they condemned someone who was later executed, then, later still, new information emerged which proved them to be innocent.
Just by the pure first-principles of logic, the death sentence is a bad idea.

Video footage in the age of LLM is completely worthless as evidence.