Capital punishment certainly works as a deterrent in the sense that a dead person commits no crime. But the way its conducted and the number of false positives

Capital punishment certainly works as a deterrent in the sense that a dead person commits no crime. But the way its conducted and the number of false positives makes me not a fan, in most cases. I imagine there are some situations where it is highly effective, but only if done immediately; I cannot fathom the whole "death row" thing for those hypotheticals. I believe strongly in innocent until proven guilty, speedy trial, due process, beyond a reasonable doubt, preponderance of evidence, trial by jury of peers, and all that fun stuff, and I acknowledge even all that goes wrong sometimes. However, assuming full uncoerced confession and irrefutable evidence and eyewitness testimony, why do we still make a (say, serial killer) suffer in death row instead of just execute them immediately? Revenge?