No need for having multiple sources to suffer parts size variations. Parts are made in batches afterwhich the same machine is reconfigured for another part or another step on the same batch. True in the old times of mechanical stops as well as in CNC of today. An error during reconfig, a larger tool wear there, temperature variations and you may be making parts significantly different. Very old watches were made practically by hand so to fit each part to another. Russians probably had decent machines but onerous production targets, impossible to impose strict tolerances so if the problem was correctable at assembly, so be it.
I've used cocktail sticks, or rather toothpicks, which are "free" from the works canteen. They work fine for general cleaning and manipulating small parts. They are not sharp enough for pegging jewels though.