... which doesn't work. And if the victim's family is allowed to observe the execution of the prisoner, for example, that has nothing to do with deterring crime. There are countless other examples, especially in how both public officials and law enforcement agents, and the media portray court cases.
All the talk about justice being served is missing the point: if punishment is meant to deter crime (either by subduing/rehabilitating the prisoner or by serving as an example to would-be criminals), the feelings of the victim or the victim's family are irrelevant.
But we can actually measure whether punishments are a useful deterrent. Harsher punishments aren't better deterrents. In fact, they may even have the opposite effect: if you're already committing a crime that incurs the practical maximum (i.e. life-long imprisonment or the death penalty), the potential punishment for further crimes can not provide any deterrent at all. What can you practically add beyond that? Torture? Mutilation? A slow and painful death?
I'm not going to argue that the complete absence of punishment would be better. But in order to serve as a useful deterrent punishments often would have to be comically disproportionate to the actual harm of the crime. And that's exactly what we have in the US.
There are exactly three possible explanations to why the US is beating the entire Western world in terms of prison population: a) every other country in the Western world is overrun by criminals because we don't lock up enough of them, b) Americans on average are much more likely to be criminals or c) American prison terms are absurdly disproportionate. I know the right-wing media (or by non-American standards: the far right-wing media) likes to promote a) and Hollywood might be sold on b), I think we both know the answer is c).