That sounds logical enough... On my "modern" single actions, however, wrestling to get the cylinder re-aligned with the cylinder pin so that the pin will engage is sometimes quite challenging. On the Colt cap-and-ball revolvers, the cylinder pin stays attached to the frame (or is it to the barrel assembly?), so that it's a fairly easy matter to align the cylinder and drop/push it onto the arbor. I'm betting that the old-timers, whichever system they owned, simply practiced and practiced until they "got good" at it, whichever system it was.
Oh darn! Look at that! I just now re-watched the beginning of this clip. That's a Remington (has the top strap) Eastwood has in that scene, not a Colt! And there are a few frames that show (albeit not too clearly) the butt end of the removed cylinder. From those few frames, it appears that it contains brass cartridges, not percussion caps.
I can see exactly what you mean by "simply pull the pin / push the pin". It is almost identical to dropping / reinserting the cylinder on a Colt SAA or a Ruger Blackhawk. Eastwood makes it look easy without even looking down, but I'm guessing that if he messed up they simply did a re-take.