I'm more comfortable with a utilitarian idea of justice, where retribution and intuitive feelings about justice being satisfied come as side effects and not explicit goals. Deterrence, prevention, and rehabilitation are the goals. Punishment needs to be shaped by those.

I would certainly not say that all functions of punishment in society are legitimate. There was just a news article about someone whose exculpatory DNA evidence will not prevent them being executed. Assuming that sort of thing happens occasionally (regardless whether it's true as reported in this case) then it certainly serves a societal purpose to avoid admitting prosecutorial error, but it's not legitimate.