I'm a huge believer in that grand old American warhorse- The .30-06 Springfield and 220Gr Remington Core Lokts (a very outdated bullet in the eyes of younger generations of sportsmen, but nevertheless one which continues to serve me well ever since my first African Safari in 1974).
If Remington Core Lokts are too old fashioned for your tastes, then I will highly recommend the 200Gr Hornady ELD-X Red Tip bullets (either hand loaded by yourself or custom loaded by a company like Hendershots Custom Ammunition). Extremely efficient modern design for all African plains game.
If you are tempted to take shots out to longer ranges, then I would highly recommend dropping the bullet weight down to 180Gr for a flatter trajectory out to greater distances (such as the open terrain on the African plains).
But 200Gr (or heavier) bullets are absolutely unrivaled for brush hunting.
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I have only shot a 7mm Remington Magnum caliber rifle a couple of times in my life. But I imagine that it would also be an excellent choice for plains game. I am continuing to experience great success with my 7x57mm Mauser Churchill Gun Makers Model Deluxe on a wide variety of international game over the years.
Mostly with the (now unfortunately discontinued) Winchester Super X factory loaded 175Gr soft point. But also more recently with the Hornady 175Gr Spire Point Interlock (as loaded by Hendershots Custom Ammunition).
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I have successfully used both calibers for making one-shot kills on all of the African plains game up the gigantic bull eland. But for eland and Kudu, the 220Gr bullets fired out of a .30-06 Springfield are more ideal for shots taken at acute angles.
All the best.