Your hunch is right - that adaptor will not work in South Africa. I have the exact same type and needed to buy special South Africa adaptors. It's the first country I'd been to where it wouldn't work.
You need one with a plug that looks like this. I've never seen a multi-country adaptor that includes this type - they might exist, but it's probably easiest to by a cheap South Africa adaptor then plug your USB plugs etc into it.
