The only things I've had kill osage bows were insect damage, water damage and my own ineptness. IMO, ring thickness just makes chasing one more stressful. I'm working on a bow now that has rings about 1/32" thick. I have one ring to go and if I botch that I'll sinew back it. The knots make it quite a bit more challenging, but unless they're really large or water has gotten in around them you can make the limb bend like normal. Sometimes they loosen up in time and you can take out the center. With osage I would always go for it. If your worried, back it. But, that old wood is so strong I doubt you have much to worry about.
I tried to get a little greedy last night and split one of the small split fence pieces in two and get 2 bows out of it. Like always, it split out the side after going about 4'. I took that split and tried to bend it against the floor, floor tillering style. I could not bend it. I tossed it in the garbage, but I may fish it out tonight and hang on to it for the next time I do that. Would like to put it with another similar split and make a really thin bow. This old wood is so stout it is amazing. Not sure how I would splice it though. After 60 years it is still yellow inside. It must've been better growing conditions back in the 40s as the growth rings on these fence posts are pretty thick. I like that.
George