While TWC’s publications use “the player” or “they” as I noted, when referring to historical generals, even generically, i.e. “the Prussian corps commander” we use “he/his” as there were no female generals during the Napoleonic Wars. Often times the Prussian corps commander at the tabletop will be controlled by a player, who could be of a gender other than male, but as our rules have specific terms that refer to a player (Force Commander is a game term generally referring to the player who commands a force), we treat the two instances differently.
If we were writing in a context where the presumption of the ‘actual’ general could be of variable gender – say a modern game, a historical period in which there were female officers, or a fantasy or Sci-Fi universe where the characters were not exclusively male, then we’d switch it up from one example to another when referring to those in-game personas.
Something to consider is that our first intent was to use ‘he’ in some examples and ‘she’ in others throughout the text. We chose to use “the player” and “they” to exclusively refer to the player because we figured that in historical wargaming we might actually irritate some potential customers if we sometimes used “she”. In a fantasy or Sci-Fi genre game where the notion of people who aren’t men participating is less explosive, we would not have hesitated. Which I suppose does insert some irony at the comment “party lines” in this debate.