I tried it just cuz.
The vacant lot I cut it from is right in the neighborhood and barely out of my way between here and work. It was a pure experiment. I was thinking about how tough it seemed to me to make bows with stone tools, and wondering how to minimize that work by doing things like notching and splitting, chiseling, etc., to rough the bow out.
What I was mostly interested in was seeing if the stave got dry enough sitting there, that the wood was stiffer than the green, live wood it was attached to, so it would pop off when stressed. But, I don't remember there being any advantage to it. The stave WAS stiffer, but it still had drying to do, still tried to warp, etc...
That same year I did the experiment where I cut the tree DOWN, but left the whole top with leaves on. I wanted to see if the foliage, plus the capillary action of the trunk would pull a lot of moisture from the wood and prevent drying issues. It sort of did. In just three or four days, that stave, with bark on, had lost a ton more weight than others cut top and bottom would have. The bark was still easy to peel, and I peeled it, and left it whole. It did warp, but didn't check, which it usually would have. It wasn't any better than roughing out a stave to dry, though.