So instead, I borrowed an electric sander from a friend and have generally finished the first chair - it probably took 2 hours total. I went through a lot more paper than I expected and even the electric sander took some patience. I've now decided that the arms will be painted black so, although I will sand into the tight corners by hand, I've just been aiming to get the varnish off and have stopped worrying about getting all of the colored stain underneath off. That is the upside of painted arms!
For lovers of these types of chairs, before you shoot me, the wood is just old pine...I'm not defacing a collectors item with black paint, I promise!