They could have gone with any popular multiplayer genre. DOTA, Starcraft, WoW, GTA, Saints Row, bloody Minecraft. Any of those would have been preferable. Or maybe a better FPS type game as the base. Like the only amazing one, "Painkiller."
FPS games are still the Football of the American world. It makes sense to have that as the sport everyone cares about the most because if eSport did get popular in the mainstream world they would likely be the most popular.
You are coming at the issue from a minority standpoint, and while I agree with you that I enjoy things like LoL and Starcraft better, FPS makes more sense in the context of the story.