Your argument is OK as far as it goes. But I believe the OP did make a point, while kind of vague, about asking who executes who if you execute an innocent man? It has happened more than once in the past. And we constantly see people exonerated who were convicted of capital murder and sentenced to death after a trial. So who do we execute if the trial finds that an innocent person should be executed and it happens? Do we execute the jury? The judge? The police? The prosecutor? The defense? Any of them? None of them? Why?
For what it's worth, I don't have too much of a problem doing away with someone for murder when it is absolutely for certain that they caught the real bad guy. But I don't like it dragging on for years, and I don't like it done humanely behind walls. Make it horrible and in public. That way everyone sees what's up. Out of sight out of mind need not apply here. And for me, if an innocent person is executed, find out why and execute the people responsible... but at least one of them has to be an official... even if it includes jury members. That would have made trials of blacks by all white juries in the south more entertaining. Let them wear their uniforms while swinging. It would look like putting out the bed sheets on the line.