But, in practice, we will never get it 100% right. Probability of killing an innocent person will always be >0.
For that reason alone, we should get rid of it.
The importance of that I think is clear with respect to the dynamics for terrorism and war. People who experience their loved ones dying easily get vengeful and radicalized.
If I lost somebody dear to me, I don't think the execution of the killer that my increased satisfaction from this would likely pale compared to the sadness inflicted on those who cared about the killer.
This is a fundamental thing in human psychology. We react more to losses than gains. If e.g. the boss takes money from you and gives to me as a bonus, then my joy from this will be significantly less, than the dissatisfaction you will feel from this loss. The sum is lesser than its parts so to speak ;-)
If you put a serial killer to death, you can be 100% sure they won't kill any more people. If you put him in prison, you can't. He may escape, or he may get released early and kill again. Or as often happens, he'll kill other prisoners whilst in prison.
It's a question of whether you value potential victims, or potential falsely convicted innocent people. IMHO the former is a far bigger problem.
We have different tolerance for accuracy depending on the context, and my assertion was that in this case 100% would be necessary for, at least me, to feel comfortable carrying out the punishment.
If you don't put them to death, can you be 100% sure saving that life isn't going to cause more people to die?
It's not simply a case of "punishment". It's preventing further crime and deaths.