The main indicator of how powerful a bow will be is the bow's draw weight. The harder the bow is to pull back, the more energy that is being put into the bow by the arm. The design and materials of the bow affect how much of this energy the bow will store, how efficient the bow will be at imparting that energy to the arrow, and what kind of draw weight can be achieved without breaking the bow.
English longbows and Mongolian composite bows both have designs with high energy storage and efficiency. Longer limbs are under less strain when being drawn and horn/sinew are strong, elastic materials, so both longbows and composite bows can be made with high draw weights.
I think one lesson we can take from history is not to under-estimate the ability of "primitive" people to organize effectively. Lately I've been reading about the Tarascan empire, which was a rival state to the Aztecs. The Tarascans migrated to Western Mexico only a few centuries before the Spanish conquest, conquered the surrounding chiefdoms, and then formed a government that was more centralized and better organized than the Aztec empire.
When the Tarascan empire was formed it was the first time a state-level society had been developed in that region. The Aztec empire (which was also a state-level society) followed the same patterns of political organization that had existed in Ancient Mexico for a millenia.
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