My Experience has been that the majority of old linen stops were pasted with a chalk/ Pipe clay based whiting. The whiting was a way to smooth the strop surface, prevent dirt from collecting in the weave, and also stiffen it.
I did some experiments with the clay formula that was used by British soldiers in the 18th century to whiten the cross belts and gaiters of their uniforms. I have a couple strops that still, after a few years, are not discernible from the pastes that you folks found.
Still, today, many of the linen components of strops are treated the same way. Like the Illinois #127 especially.
JME, YMMV
~Richard