I think this is a fascinating topic. But since there is such a lack of archeological evidence with regards to wood artifacts, who knows if we will ever know the full story.
I think there is a "eastern woodland bow" that is a generic style. And I don't think it does any dis-service to any culture, because I believe it represents a utility bow if nothing else. A feed your family bow. I think it has to be made with the woods of the east to qualify as a eastern woodland bow. If it is made from yew then it becomes an english long bow. The people who made these bows in the past had good reasons to do what they did, and made the best of their local resources. There is no reason to think that the people who lived in the wide area of the "eastern woodland" would not share their ideas, and if making a long, narrow, bend-in-the-handle bow, made sense, why wouldn't they do it? Especially if it could be made fast and easily. Imagine if making a bow was a chore, not a privilege or a hobby. Wouldn't that lead to a simplified design. And if someone was talented and innovative they would add some carving, or make a double bow or otherwise add something to the design to make it "theirs". Or a certain unique piece of wood may lead to a special design, like our character bows.