There are many twists and turns to this. For example, few woods make suitable bows, and even then, most can only be used to make “short bows”. If the bows are too long, they will break.
The exception is the Welsh or English longbow, made of Yew, that first made its appearance known in the 7th Century, but was still a strategic advantage over the short bow in the Battle of Agincourt (1415), best known for the Shakespeare play, Henry V.
Unfortunately for those in warmer and drier climates, though they obtained Yew bows, the wood soon dried out and they broke in ordinary use.
Back to primitive times, the use of bows and arrows was quite different on an individual basis, compared to the tactics of a group. With spears, the emphasis on the group was, for example, a single animal. But arrows were so efficient that as soon as an animal was down, the group could concentrate fire on another animal.
In turn, this forced some degree of nomadic behavior, as once animals were killed, they had to be processed on the spot before they rotted. This included tanning hides, smoking and grinding meat, carving bones into tools, etc.
And with human conflicts involving bows and arrows, spears and other weapons still played a major part.
My copy of "The Traditional Boyer" begs to differ with you. They've made successful hunting bows with just about any kind of wood you care to name: Oak, elm, maple, hickory, ash, osage orange, even mulberry. I was very surprised by one of their tests showing a 6' maple bow launching arrows with greater force than a yew bow of similar length.
Another thing that surprised me was that one of the oldest bow fragments they found in England (going back a few thousand years, if I recall correctly) was about 6 ft. long, but it's limbs were flat, rather than the "D" shape of the traditional English (Welsh!) long bow. Apparently, the flat profile of the limbs lets them move more efficiently and impart greater speed to the arrow.