It just takes a combination of patience and remembering that capillary action is not just your enemy it is also your friend. When done right you can actually draw stains out of the paper to a degree same with toning. To be clear though it is not strictly chemical free there is a cleaning agent, a type of soap compound for lack of a better term, used just not bleach. The cleaning of this one before backing was done over a period of days not hours.
I have much better documented in pics the restoration of the River Girl one sheet I have which was worse than this including dirt, staining, stains from paint, tape front and back, etc. Will post it in one of the resto threads if I ever finish it off. That one will have some paper fill done though.