We've killed two previous Dons. The current don will be an old regular member who was chose nto be the new Don. The original Don's "power" is to appear good when investigated, and then he gets the extra responsibility of chosing who he sends to kill who. Once he dies, and a new don is elected, his "power" remains the same. He doesn't gain the Don's ability of appearing good, only the extra responsibility.
I know this from a previous game when I was don. Its possible Moogle plays it differently to chairdriver did in that game.