What if the sentenced person is actually innocent? No amount of apologies or recompensation will bring that person back.
But at least the question "how is that acceptable?" is in fact a question of a moral nature. It's unacceptable, but it is unacceptable because it is immoral.
While I don't like the death penalty I don't think it's that different from a very long sentence. I don't think it makes sense to say that any punishment needs an absolutely perfect error rate.
Advocacy in favor of the death penalty is never about "death penalty for murderers/rapists" but "death penalty for people convicted of being murderers/rapists". Practice has shown there's a big difference
That's a pretty big step, and to me it requires a lot of benefits to justify.
It seems to me like taking care of business before that happens is a more beneficial thing.