But the only way to prevent a miscarriage of justice is to have no justice system.
The only way to
absolutely prevent
any miscarriage of justice is to have no justice system, yes.
We pretty much all agree that a justice system is necessary however, and the posiiblity of miscarriage of justice is why we insist on due process, safeguarding the rights of the accused and placing the burden of proof on the prosecution.
We also make it part of the system that if new evidence comes to light exonerating a convicted person, we let them go. We do this because we care about justice.
The death penalty specifically denies the opportunity for further evidence. It's proponents either deny the possibility of miscarriage of justice or think killing prisoners is so desirable that they just don't care.
Having the death penalty is a statement that at some point you are just going to stop looking at evidence. You've seen enough. Nothing will ever make you change your mind about a persons guilt. It's a statement of faith.
Sure, most of the time you will be right. Eventually you won't be.
The death penalty is irrevocable and specifically denies a prisoner relief or redress in the event that evidence exonerating them comes to light after their execution. As long as even the theoretical possibilty exists that the wrong person could be convicted of a capital crime the death penalty is incompatible with justice.
Given the highly controversial nature of some real world capital convictions it's more than just a theoretical possibilty. Given the well known flaws demonstrated by any justice system, it's an eventual certainty.
ETA: Oh, and the State killing an innocent person is pretty much the most morally horrifying thing that could be done in our name. If all we need to do to avoid this is
nothing - ie refrain from executing prisoners in our custody, the whole thing seems like a no brainer.