Because that would be boring AF. FPS games are primarily about a single player experience, which is usually relatively fast-paced. Civil War era rifles are slow to load and tended to be fired in formations akin to those predominant in the Napoleonic wars. Who wants to play a guy marching in formation whose top speed is 3 rounds a minute? (Maybe less, due to the rifling.) And it wouldn't make sense to have a single player running around on their own, firing off cartridges that would take, like, 40 seconds to reload just for a single shot.
This is why the earliest era you tend to see in FPS gaming is the Wild West -- more repeating weapons like a lever-action rifle or a revolver. (Yes, I know the Civil War had revolvers, but it didn't have many repeating rifles in wide circulation, if any.)