I don't like immortality. Limited mortality for important characters is a positive thing, but if a champion can never die, that's silly and robs us of a goodly portion of the thrill of combat.
They should be free agents after the defeat of their empire, sure. Champions should also be able to defect, but that should be very rare, only a couple times in a long game. That way when it does happen it's more compelling. This could also be related to a captive system; captivated champions could be given an opportunity to defect in addition to things like interrogation, ransoming, and execution.
I like Romance of the Three Kingdoms stuff. That is probably a post-FE thing though, since relationship frameworks are essentially a more broad form of dynasties, and making dynasties interesting is a post-FE goal. But yeah, personal friendships and rivalries would be great. Personality traits could probably work passable-fine in the current framework, though of course I don't know enough about it to say definitively.
New rulers aren't chosen in FE, but there's still interesting gameplay that could exist regarding that. If Stardock does choose to go down that road (which I think they ought to, turning champions into interesting characters should be a priority in my opinion), then they could have us appointing a champion as governor of a city. Masters of Orion did this well, but in FE's case the governor would be a character who can move about on the main map and fight, and who is a part of other friend/rival and dynasty systems. Potentially there could be penalties (or lessened bonuses, rather) associated with keeping a city's governor away for too long.