I wouldn't exclude the "Big Five." John Kingsley-Heath, who may have been the most experienced professional leopard hunter in East Africa, considered the .243 Winchester ideal for baited leopards.
The African PH I know best, the now-retired PH I hunted with most and know best, killed 500 Cape buffalo including many big bulls, with the .30-06 and 180-grain Nosler Partitions, during "culling" drives on a big ranch in what was then Rhodesia. Never had a problem, and any ballistic difference between the .308 and .30-06 would be only vaguely theoretical.
Dunno how many "huntable species" there are in the world, but the Weatherby Award winners have generally killed around 200. Some weigh 20 pounds, or even less. Have seen and photographed one of the subspecies of dik-dik in Nambia, and they aren't any bigger than Montana white-tailed jackrabbits.