Plenty of buffalo were shot in the old days with 303's, 6.5 Mannlichers, 7mm and 8mm Mausers with open sights. All tended to use long for calibre round nosed bullets that penetrated well. But a lot of this was n't singling out an old bull and tracking him etc etc, but much more meat hunting for rations. Big Game rifles were expensive and ammo just as much so - they were rifles for Safari Hunters, not for farmers and settlers etc who would just have one rifle they shot everything with and the cheapest were military based bolt actions - Lee Enfields, Mannlicher Schoeneurs and Mausers.
The 375H&H, 416 Rigby etc were unaffordable to most settlers.