>This seems to be a very recent thing. First off, I don't think the justice system is meant to "bring closure" to the victims,
A stable society needs a mechanism to achieve justice other than vigilantism. So yes, "closure" (and, as mentioned elsewhere, "revenge") are aspects to punishment.
There are several aspects to punishment that must be considered to have an effective justice system:
1. Deterrent - the punishment combined with the chance of getting caught should ideally cause most people to reject committing the act in the first place.
2. The punishment should protect other people from the person who has shown willingness to cause harm.
3. The punishment should be severe enough as well as proportionate to provide a sense of justice/fairness/revenge to the victims and remove the impetus for vigilantism.
4. Many people also want the punishment to rehabilitate the convicted so that they no longer pose a threat to others.