I don't know about self-inserts, but personally I feel like a lot of the appeal of a slave trainer comes from the unfamiliarity aspect. Taking a complete stranger and molding them into a whore. I think 4 elements has way bigger problems than Aang, but it being Aang just takes me out of the action. Feels like there are expectations as to Aang's behavior and how people behave around him. When those expectations are met, it's not suitable for a slave-training porn game, and when those expectations aren't met, it creates cognitive dissonance.
I think if you're doing a slave trainer it should basically be mandatory to have the player play an OC. Not for self-insert purposes - you can give the OC a very strong personality and appearance, make them impossible to self-insert into. Just don't make it Aang or Dumbledore or Genie or whoever, because your fanfic-tier personality for them is going to come into dramatic conflict with the player's preconceived canon personality for them. And in the end, it's just sexier when you're some nobody who manages to break a princess than when you're the Avatar.