Yes that's usually the motive behind its supporters but not the used argument when it is criticized.

Yeah. People (particularly politicians, in public) are afraid to say that mostly what they want is vengeance.

So they frame it as "deterring others in the future" which sounds sort of noble.

It's like they know the vengeance thing is f--ed up. Why else would they lie?

Perhaps even worse is that the politicians themselves might not even crave vengeance. A lot of the time, they're just pandering to voters who want it.