Head-Smashed-In Buffalo Jump is north of me in Alberta Canada, and the Madison Buffalo Jump is over the hump by Bozeman, MT... It was a common practice.
Folks forget that the horse was a very recent addition to Native culture in the Americas (and I wonder about the bow too) Without such things,(and even with them as the Jumps prove) it is easier, and far less risky to herd an animal over a cliff, or into a squeeze chute in order to dispatch it without risk - I think that such practices were naturally far more prevalent.
Every hunter (including man) is an opportunist, with risk necessarily lent more weight than anything else.
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Pits have also been used to trap game. It wouldn't surprise me if many of these techniques have been used since the dawn of man.