Im am not very experienced with osage, but with every other wood I have worked ring thickness made no difference by itself, except that on some very thin ringed locust I had trouble chasing down to just one.
What matters more is the mass/volume of the wood. heavier wood and lighter wood regardless of species need to be treated differently. And relatedly, the ratio of early to late wood. Some rings are so thin that the thickness of early wood is about the same, and it gets pretty porous and brittle. In Utah, I can see our 20 year drought in the outer rings of locust and mulberery trees. I'm having to cut down to the late 90's to find a good back ring.