Zach, the 375HH is the most ideal cartridge period. It will take anything alive and not blow a 20 pound duiker to bits. It will shoot a buffalo at 50 yars and kill him easily with a well placed shot. It will take a kudu or gemsbok at 300 yards if needed. It's flatter shooting then a 30/06 when used with 270 grain bullets.
Of the three you listed the 7mm mag, 30/06 and the 300WSM the most logical choice is the 30/06. The others will blow to bits the smaller game. They are just too fast at close range and must be used with premium bullets for consistant performace at any range.
The 30/06 is by a large margin the most frequently used gun by Local South Africans. The 300 mag is the most common gun brought by American hunters. So it would appear that the .308 diameter is a good choice. I have used my 30/06 there and taken nealry every species of plains game with it. Never lost a single animal. It stuggles on the bigger stuff once in a while but it works when you do your part.
The vast majority of shots on true wild game in the bush will come at 150 yards or less. So the magnums are not a perfect choice. They are better for the open areas of teh freestate or the Kalahari, but a poor choice in the northern province, or zululand. If you have to bring a magnum I always suggest the heaviest bullets your gun shoots well and then make them bonded core if possible. Hitting a big animal at 40 yards with a 3000 fps standard bullet is gonna be real tough on that bullet!
The perfect plains game rifles are the 338/06, 338, 375, and 35 calibers like the whelen are good too. After that the 30/06 is hard to beat with 180-200 grain bullets.