In such cases the Death Penalty is rarely, if ever, used. There's Life Imprisonment for those cases with shreads of doubt, and it's easy to distinguish the ones with and without a shadow of a doubt.Sorry, i misread what you were saying. But in cases like you're describing the process usually is much swifter and less costly, but those cases where the evidence is incontrovertible are very rare. People have been exonerated after being given the death sentence; if it wasn't for the time between the sentence and the execution they would have been murdered for a crime they didn't commit, and no matter how rare those instances are as well they cannot just be dismissed for the sake of saving money.
The system fails the other way around too. Meet O.J Simpson - the guy who killed his wife and her lover, got away with it and practically admitted it in his autobiography.If you're suggesting to implement a system where you can execute a criminal immediately after conviction, even if it's "obvious", there is going to be a LOT of abuse.
The system fails a lot as it is. Don't make it even more exploitable.
Not being able to take a person to court once the highest authority found said person not guilty even after new evidence surfaces is another cancer on your judicial system you should take care of, but I digress. My point is - no system is perfect, but some are more efficient than others. Yours isn't efficient - it costs too much, it overcrowds your prisons and it's filled with procedures that are not always necessary.
I'm not saying that a Death Penalty should be carried out due to circumstancial evidence. I'm saying that when there is no doubt, there is also no reason at all to wait.