While something like the 280 Rem will work for the animals you've listed (I have taken all 3 with a 7mm Mag) the caliber is on the light side just like it was when I did it. Both zebra and gemsbok are tough animals. Impala are not. My recommendation is to use a bigger caliber than a .284. However if you insist on using it, use very good bullets and be sure you place them in the right spot. Zebra are notoriously difficult to place a bullet on since the stripes seem to through people's aim off. See which one of the premium bullets your rifle likes and use them. However there aren't really a lot of choices in the .280 Rem and you didn't say if you handload or not.

One problem with something like the 280 Rem is that if your rifle shows up and the ammo doesn't (this did happen to me once) you may not be able to find any where you are. Some of the more unusual rounds aren't easy to feed in many places in Africa. South Africa is a pretty big place so you make be in an area where there are few gun stores. That's why many people take common rounds such as 375 H&H and 30-06/270 Win/7mm Rem Mag on safari. You can get ammo just about anywhere for those calibers. That is always something to consider.