Arachnid, you are right about the board ending up thin on one end. The cure is to look for a flat sawn board where you can see good rings, and one ring, barely exposed, running full length down the face of the board. the rest of the growth lines will run parallel on each side of the middle. I once found anred oak board like that with a SINGLE growth ring half an inch thick making up most of the board. Best oak bow ever!
BTW, I have made some pretty passable bows from tool handles. Two ash shovel handles will be top quality white or European ash, have visible growth rings, and you can tell whether the grain runs off the side with a practiced eye. That'll make a really good American longbow up to 7 feet long. Chase a ring and socket, or splice a handle and leave it thick. Same thing with hickory ax or pick handles.