I certainly love the idea of the government, who has our best interest at heart all the time, and our legal system, which is perfect and never subject to bias and emotion, having the ability to swiftly just kill people. What can go wrong?

Also, that's literally the same problem. Once they've committed a crime, they've no longer got anything to lose since you can't kill them twice, so now you've incentivized them to just go balls out on horror while they can.