Fin:
I too am taking my first safari this fall, but, I'm taking a different perspective. To be blunt, you]re looking for an excuse to purchase a 375H&H, which is fine. At the same time you have to be realistic. On your list of animals I didn't see an Eland mentioned. THe Eland is the only plainsgame that would justify a 375 H&H. All the other antelope can be had with your generic Elk rifle. I'm taking a 270WSM for the bulk of my hunting. I'm also taking a 338 Federal and the only reason is for my Eland. All the other animals can be taken with a 30-06 without issue. They don't need a 375 H&H or even 300 Win Mag.
That being said, if you need an excuse to purchase a new rifle, Africa is a great reason to go shopping. Personally speaking, I've shot my 270WSM for years and am very comfortable with it. At the end of the day we all know bullet placement is what matters and I can achieve that with my 270WSM. I'm leaving my 375 H&H at home because I don't need it. The 338Federal will be enough rifle to get the job done.
Enjoy your hunt and enjoy the months spent planning and organizing for it, I am!