Jmar,
You can read about Confederate irregular forces, who made up for the lack of breech-loading repeaters with multiple revolvers. It was not uncommon to see many of them with two, three, or four revolvers, either on their person or in pommel holsters. As opposed to multiple cylinders, the use of multiple revolvers is well documented. I imagine that a number of those were older, pre-war revolvers, or battlefield pickups.. As for regular troops, if they were at pistol range and emptied their pistols then they'd tend to draw a saber or withdraw. I personally have trouble imagining a man having an easy time of swapping cylinders whilst on a moving horse in the middle of a firefight.
As for the Remington not being that common, well over 200,00 were made during the war (the second most produced). It was certainly a common enough firearm during the conflict.
OP, sorry to get off topic, absolutely my bad.