I'm not very experienced or knowledgeable about pistols, so maybe this method of using the offhand to pull the hammer of a pistol back, rather than pulling the trigger, has some utility that is not obvious.
But, it seems as though it would jostle the weapon more (reducing aim), and also be slower, than simply using the trigger.
Is there any evidence that this method was actually used? If so, why? Thank you!
Others are going to be more qualified to speak on whether fanning actually happened or not (I think it was more of a party trick than an actual gunfight thing), but I can at least explain why you couldn't just pull the trigger over and over to fire quickly like you can today. Old revolvers were what's called "single action", where the hammer needed to be pulled back after each shot, which rotated the cylinder and made it ready to fire. Pulling the trigger just dropped the hammer, it didn't rotate the cylinder or re-cock the weapon. Modern revolvers are "double action" which means that while you can pull back the hammer, it's not necessary as pulling the trigger itself pulls back the hammer, rotates the cylinder, and then drops the hammer on the new cartridge.