Another aspect that I haven't seen much discussion of, is: punishment is there partly to stop other people from exacting it.
Situation one: A kills B, B's brother kills A, A's brother kills B's brother... and eventually A's family and friends and B's family and friends are just killing other all the time.
Situation two: A kills B, the state kills A. Done. B's family has nobody left to kill, and A's family isn't likely to try to fight the state.
This suggests that the state can't make punishments too lenient, or people will take matters into their own hands again. It's not stable.
Yes, in exchange for giving the state a monopoly on violence, people need reassuring that the state is going to dispense an appropriate amount of it when required.
Situation one: A kills B, B's brother kills A, A's brother kills B's brother... and eventually A's family and friends and B's family and friends are just killing other all the time.
Situation two: A kills B, the state kills A. Done. B's family has nobody left to kill, and A's family isn't likely to try to fight the state.
This suggests that the state can't make punishments too lenient, or people will take matters into their own hands again. It's not stable.