In the same way that some people argue that the death penalty applies too much physical suffering, there are those who claim that the murderer suffers too little, compared to his/her victims. But they’re looking only at the physical aspects of it. Arguably, the greatest agony of the death penalty is mental and psychological - the suspense of it - not physical.
Many, perhaps most, of the people freed from death row by overturned convictions were freed by evidence not available at the time of trial or initial appeal–DNA evidence that had never been processed, for example, or for which the technology did not yet exist. How would your hurry-up tribunal address this problem?
Beyond that, a lot of the issue is very long delays in state appellate court. For example, the notorious killer Jonathan Carr (see Wichita Massacre) docketed his appeal in the Kansas Supreme Court in February 2003; the court produced an opinion eleven years later, in July 2014. Most of that time was not waiting on judges to rule; it was waiting on lawyers to file briefs. His lawyers took six years to file their initial brief; the state took three years to file a response, then there were a couple of rounds of reply and supplemental briefs before the case ever got to oral argument. (The case then went to SCOTUS and is now back in the Kansas Supreme Court on a motion for ruling under state law; the most recent round of supplemental briefs was filed yesterday morning.)
I knew an Indian man in Thailand. He told me about this one condemned prisoner back where he came from. Said they simply forgot to execute him, and for the next 15 years, the condemned prisoner lived in dread that someone was going to discover the error. Sure enough, one day while going over the books, the mistake was realized. So they immediately took the prisoner out and hanged him.
He gave that an example of why he preferred living in Thailand. Thailand’s crazy, but India is positively insane.