The .30-06 won't just "do in a pinch" on plains game, it will flat do.
May not have shot as many plains game animals as RinB (who I know well), but have been month-long cull/meat hunters with a lot of other people, so have witnessed far more hunters in action (and cartridges, and bullets) than just those used by me. In general, the .30-06 is more effective than larger cartridges (even with such "antique" bullets as the Nosler Partition) because it doesn't normally result in a flinch during a 10-14 day safari, unlike harder-kicking cartridges. Here ist should be mentioned that you probably won't be shooting just at game, but also "check-shooting" the rifle every 2-3 days, to make sure the scope is still on after bouncing around in a Land Cruiser.
Have hunted with people who can shoot harder-kicking rounds consistently on such a safari, but they are in the minority. Finn Aagaard was of the same opinion, estimating that only about a third of his clients could consistently shoot .300 magnums accurately. Part of the reason is that the typical safari client tends to be older, because that's when they can afford it, and consequently may not be quite as recoil-resistant as they were when younger.
There's another factor: Cartridges harder-kicking than the .30-06 also tend to be tougher on scopes. Have seen plenty of previously reliable scopes, costing up to well over $1000, fail on a safari, not just because of bouncing around in a Land Cruiser but because the rifles they were on really hadn't been shot all that much beforehand. This was because the rifles kicked a little too much for their owners to really practice--which also means really proving the scope.
