The pizzas turned out quite well.
The proofing in the wooden box was much better than in a plastic dough box. No 'sweating' of the dough, no drying out, it kept a nice and smooth, almost silky, texture, no sticking to the bottom and no 'skin' forming whatsoever. I left a couple of dough balls in the box overnight at RT and when I checked this morning, I didn't see any changes at all, except a bit more expansion.
I think a wooden proofing box is a pretty good tool.
I don't know about drawers. I could imagine that they are sometimes treated with some sort of laquer or other chemical.