The perfect system is one that leaves open the possibility that society will be able to correct its mistake. That really is not a perfect system. But it is a whole lot better than the one we have now.
You would have a point if we were executing criminals within minutes of their death sentence being handed down. As it is, if the perp, his lawyer, and the legions and anti-death penalty activists and lawyers can't convince anyone he is innocent for a dozen years, the chance that he is innocent is too small to be considered seriously.
The average death row inmate will spend at least a dozen years pleading his case, and has an excellent chance of getting it thrown out anytime a supreme court justice wants to impress his cocktail party guests with his amazing compassion for murderers. Never mind his guilt.