And if no one ever determines that the innocent person is innocent, then their life is completely wasted in prison, in my opinion.
New technology like DNA, deathbed confessions, evidence found or witness statements withdrawn years or decades later can show a ruling, no matter how correct it seemed at the time to be wrong and without there being any active investigation. Sure if I've been ensnared by unfortunate circumstances or framed I'd rather you find the truth straight away, but I'd rather be wrongfully imprisoned than wrongfully executed. As long as there's life there's hope that I'll be a free man again and you can't conclusively say it won't happen until I'm dead.
Sure, it almost certainly won't happen but proponents of the death sentence is using the likely outcome to justify the means. It's like basing a warrant on the assumption that you'll find something to justify the search. Yes you've lost the presumption of innocence, but when humans make decisions on worldly evidence and testimony there'll always be a smidgen of doubt left. Posthumously clearing a name might not matter much to the dead, but it matters to friends and family and helps prove the system isn't perfect. And though it can't get better it won't be perfect and we can't turn back time, but we can give the innocent every chance they can get. And that ought to be enough.