I cannot seem to find the photos of the crossbow with that kind of wavy grain on the tiller that made me think of maple or limewood. So just forget about my idea. I also talked to my friend whose special collection I posted,
and he testified that all Gothic crossbow tillers he had ever seen were of fruitwood, mostly pear and cherry, including the three fine pieces he owns.
The white tillers you mentioned might have remained 'in the white', unvarnished, as many sources of period artwork (paintings) indicate, as well as the stocks of many early arquebuses. I'm also afraid none of those seems have survived, maybe partly due to later cleaning or restoring measures - or simply to hundreds of years of exposition to daylight.
It's so good to have a member with your range of experience here on the forum - so WELCOME!