First step towards a repaint; get all the parts to be painted off the machine.
It really helps that this is not the first Remington Portable #2 typewriter I've tried to take apart. It means that by now I know what screws to undo and when (having explored all possible wrong options on the other machines first). When you know what to unscrew, it comes apart quite easily.
Important preparation is to put drops of oil on all the bodywork screws and especially on the platen-knob screw. Leaving this to creep in for a couple of days helps to loosen. On this machine it then all came apart without any mishaps. Platen came off readily. Removing the stop-screw from the carriage base of the machine allows the carriage to slide off easily. The one item that had me stumped on the previous attempts jumped out clearly now; remove two pins (nails) at the bottom of the removed carriage and the paper tray drops off.
This leaves the core of the machine exposed for a good cleaning. The mechanism is already surprisingly clean and rust-free, considering the general state the machine was in.
Now to explore how to remove the old paint, almost certainly to be nitro-cellulose lacquer. Will be giving this a try with some acetone...