I don't think the execution would have added anything. The fact that Vimes didn't butcher Carcer on the spot, but handed him over for trial is right and proper. One of the themes of the book is sticking to the meaning of the law and not using it for personal gain. One of the lessons Sam tried to drive into young Sam was that the law comes above whoever happens to be the ruler and whatever their personal goals are.
So Sam Vimes did his job. He caught his man and handed him over to be judged and ultimately executed. His role as policeman ended at that point.
When you are watching a TV cop show, you see the cops catch the bad guy and that's it (generally). We don't see that bad guy in court or sentenced, or what happens to them in prison, because that's nothing to do with the cop show.
“Men never commit evil so fully and joyfully as when they do it for religious convictions.” – Blaise Pascal