On African plains game hunts I've used two 7mm Rem mags, a .375 RUM, and a .300 Weatherby. All worked great.
On my first trip to South Africa, I took a 7mm RM shooting 140 grain Ballistic tip bullets. It easily made one shot kills on all of the animals that the OP listed, plus a few more.
I took my .375 RUM shooting 300 grain TSX bullets and a different 7mm RM shooting 160 grain Accubonds on my second trip. I used the .375 RUM for buffalo and bushbuck in Zimbabwe, and both rifles for PG on farms in South Africa's Free State and Kwa-Zulu Natal. One of my PH's in KZN, who also had a 7 mm RM, was continually impressed with the internal damage that the 160 grain Accubonds did on the animals that I shot with them.
My third African hunt was to the Eastern Cape of South Africa just for plains game. I hunted 5 different farms or conservancies on that trip, shooting my .375 RUM and 270 grain TSX bullets. This combination was more than adequate for some little critters like a Jackal, a couple of Steenboks, and a Vaal Rhebok, and it also had no difficulty making a one shot kill on a Gemsbok at 348 yards, which is the longest shot that I've ever killed an animal at.
I did borrow a .308 Win shooting FMJ bullets from one of the outfitters for my Cape Grysbok.
I took my .300 Weatherby shooting 168 grain TSX bullets on my fourth African hunt that was in the Limpopo Province of South Africa. This combination worked great on a variety of PG from Klipspringer to Sable. I also loaded some 150 grain FMJ bullets for some smaller animals that I wanted, and I borrowed a .22 LR from my PH for a Caracal.
My last two trips to Africa were to Mozambique, and for both trips I took my .300 Weatherby shooting 168 grain TTSX bullets.
