I do not agree that, mistakenly, executing innocent persons promotes the greater good of society.
I didn't say that, of course.
Of course you did.
I am talking about those "unfortunate" lottery losers who could warrant execution (for sake of the greater society) by doing things to make life even more unfortunate for others, e.g. those who have performed, and will likely continue to perform (given the chance) the most atrocious acts of violence against others, including murder.
OK.
In that case I don't see what your objections to the penalty are. 1) we could do the killing humanely if we wanted to (interesting that we don't)
And if we sometimes killed an innocent person it would still be for the good of greater society overall.