I like linking this post by Jon Peterson because the article it discusses is actually about a group that did play the war-gaming way (with referee separate from the opposition). I actually tried something like it as a conceit in my homebrew system, but it confused my players. They thought they’d be taking on the monsters (rather than its being used as a framing device for when the referee is toggling between “neutral referee” and “playing opposition”). I’ve since reverted back to the conflict resolution process I’d been using for a while.I believe referee is merely a holdover from a time when people did not yet have a proper name for gamemasters and helped themselves with the next best thing that was somewhat familiar.
RPGs grew out of mini-games within wargames. And in wargames, you often do have an actual referee. Two players, or two teams of players are playing against each other, controlling all the units that are in the game. And the game can have an actual referee who is not participating in the play at all and not controlling any of the units, but simply making neutral, disinterested judgement calls on whether the moves that the players want to make are within the rules of the game.
Once the Special Agent Doing Spying And Sabotage minigame appeared, it was logical to have the decisions what these special agent units would learn or could accomplish made by the same person who was also refereeing the battles. At that point, the role was no longer "just the referee", but still primarily the referee. And once people were doing special mission play without the battles, the role no longer included any refereeing. But as these games where played by people who mostly had been playing wargames, and been calling the people who set up the battlefield and make judgement calls referees for decades, many still continued calling them referees. Out of habit.
But a gamemaster running an RPG is not doing any refereeing. A GM is an active participant in the play. Continuing the use of the term referee in discussions about RPGs is misleading, and I would even go as far as to say factually wrong.

D&D in the News (1976): The Duke and the Evil Balrog
Early press about D&D rarely has the luxury of wading deep into the play of ongoing campaigns. That is what makes this piece by Philip Hi...