I highly doubt bow and arrows have to do with this depletion, Australian Aboriginals were highly highly effective hunters with spears woomeras and boomerangs and there was an abundance of wild life around in their presence.
FWIW, quoting https://en.wikipedia.org/wiki/English_longbow : "The trade of yew wood to England for longbows was such that it depleted the stocks of yew over a huge area."
My proof is that they're the oldest surviving civilization on earth and lived mostly a hunter gatherer lifestyle without running out of food and requiring mass scale agriculture to survive.