I have not been to Southern Africa myself, so the big thorns etc I can't comment on but I do have some experience of sand. On the Range Rover we have the Insa Turbo Mountains. Seems like a very good all-round tyre to us and hacked through the sand quite nicely - and well priced. I dare say the tread is aggressive enough that if you did come across mud it would not struggle. If you havn't done thick sand before, remember to lower the tyre pressure by quite a bit - makes such a huge difference. Makes the difference between coasting over the sand as opposed to cutting through and getting bogged down. Wasn't really a problem in the Series 2A because of it's light weight, but the Range Rover really struggled until pressures were lowered. Downside is that you are a lot more cautious of rocks.
TBH I don't think tyre choice will make that much difference though if just sand and gravel - I have seen Nissan Micras go through deep sand and gravel on road tyres for hundreds of miles without getting stuck. In the Landie we used an old set of Colway MTs which were a crap choice really but came with the car. Drifting about in the sand is fun anyway! Did quite a few thousand miles in sand (and mud, gravel etc) on Camac ATs and it's all pretty much the same! Knowing the tyres limitations is an aspect I guess.
On another note, most of the locals and army hacking across the Saharan regions we went through use Michelin XL so they have to be worth looking at. No doubt quite a few will be using them in South Africa and will probably be readily available over there.