If an innocent person dies in prison (whether of natural causes, assault etc.), they cannot be compensated. Likewise if they serve their sentence, die, and are then cleared. You're claiming that this is unfortunate, but that it's not intended and therefore okay. However, when innocent people are sentenced to death it is not the intention of the U.S. justice system (JS) to intentionally sentence innocent people to death either.
To say that in the case of the death penalty the JS intends to kill convicts, and that when sentencing convicts to prison it does not, seems to me to highlight nothing other than the difference between to two sentences as sentences, which I have already acknowledged as a fairly obvious distinction. However, I don't see how this 'intention' has any bearing on the competency of the JS or its capacity to 'correct' or 'atone for' mistakes. If you want to make an entirely new argument based on this distinction (not sure I can see where it's going, but hey), you are free to do so, but as far as I can see it doesn't have any bearing on what we were talking about before.