What you say makes no sense. Green, alive wood is completely different and it's not logical at all to escalate that into a bow.

I'm into archery and have tried making a bow myself, using metal tools. It was incredibly hard and didn't result in something that could be used to hunt real game. Imagine doing that with stone tools and no previous knowledge.

And no, bows have no prototype stage. Unless the bow was something else and someone realized that it could be used to launch projectiles, which personally is my favorite theory.