Why do they have to be a prisoner, though? I get that some people got these types of jobs as part of a "deal" (They get their crimes forgiven, in exchange for doing a job most might not want to do.) but why can't it just be a position you can assign anyone to? I imagine if someone is "Wrathful" "Sadistic" etc. they'll take the job no questions asked. (heck, a sadistic character would probably pay you for the position of being able to execute people.)
How does the executioner work in-game? Is it a special "Publicly execute" interaction, or is it simply a bonus to execution in general? Is there any extra dread gain?
Yep, there's a publicly execute option!
Essentially the role of the executioner in society varied wildly in each culture and in different eras even. A commonly recurring theme though is the idea of the position of executioner being given out as amnesty for crimes, and very rarely would anyone opt-into the profession unless they were the children of an executioner themselves in which case they were generally already outcast.
A bit of a common misconception people have is the idea that the executioner would hide their identity, which as far as I could tell didn't really happen within our period. Everyone knew who the executioner was, and wearing a hood or a mask would have been pointless.
While you could argue a vengeful or sadistic character may enjoy torture and execution, they may not want the social stain that goes with it.