IME working with men on death row they don't actually all claim to be innocent by a long shot. I don't know if it's only in the cases where there's overwhelming evidence that they don't bother or what, but in the time I was doing that work, only one client ever claimed to be innocent and I know for a fact that he was innocent because it was another client of mine who killed the victim in his case.
I think the death penalty supporters are okay with killing a few innocent guys here and there. I don't think they ever argue that the system is perfect. Everyone knows damn well that there are plenty of mistakes made and even since people started being exonerated with DNA evidence there have been terrible legal reforms that have made it much harder for death row inmates to have the facts of their cases reconsidered after their conviction is final. Our system is more concerned with the finality of the verdict than it is with getting it right.
It's only very marginalized people that end up on death row (or, for the most part, prison at all) wrongly anyway. My client who was innocent was shockingly poor and very low IQ. I believe that most of the players responsible for his conviction (sheriff, prosecutor, judge, etc) knew it wasn't him but wanted the case closed before the upcoming elections. It was SO obvious that this other guy who was charged with murdering someone else four months later was the one. Like, painfully obvious.