You’re describing “beyond reasonable doubt”. There still exist “unreasonable” doubts, such as, there’s a conspiracy against this suspect which the entire police force, the judge and the jury are part of. Or “aliens did it”, or anything.
You might think I’m being pedantic here, but being pedantic about language is a lawyer’s bread and butter. The problem is that “reasonable” is open to interpretation, and that’s the actual reason innocent people have been put to death…
There’s no way, weird as it may sound, to definitively prove anything except mathematical expressions, it’s a fact of life. That’s why gravity is just a theory. It only takes one piece of evidence going the other way and it’s proved wrong, just like in cases where the judge, jury and everyone else were so certain of guilt that they convicted someone to death, only to find out later they should have acquitted. It’s not their fault, they were acting on the best information available to them. But it’s impossible to be sure.
That, for me, is enough to render the death penalty unworkable. It would be nice to be able to delete the worst people in society, but it’s a fantasy. It’s just not possible to do it without sacrificing innocent people on the way.