I see government killing as a compromise between personal vengeance and human decency. If the family of the victim wants vengeance on a killer, the state steps in and exacts it for them, in a relatively humane way. Left to their own devices, an aggrieved parent or spouse might well torture the killer prior to executing him, but the state has no personal interest in making the death especially gruesome or terrible.
Plus, it goes without saying everywhere save in this thread (since the OP asks that we assume "no errors") that a grieving family member is more likely to kill the wrong person hastily than the state after it's long drawn out process.
Again, I argue it is immoral for the state to ask someone to kill someone else as part of your job. If you are an individual and seek vengence, it would be much more "satisfying" to do it yourself...and then face the punishments for your own offenses. It is rediculous to have a system to "satisfy" people. It is law and order, rules that should be followed by the government and the people.