Do split beans need to be thrown away? I understand that the primary goal of sorting through dry beans is to remove things like pebbles. I very rarely find any of those, but there are often clean halves of beans. Do they they need to be thrown away too?
Nope, it's fine to leave split beans in to cook with the rest of the beans. Split beans are common, and I've never heard of anyone removing them.

They will probably overcook while the rest of the beans get just cooked, but it won't be noticeable.