Do yourself a favour . Think very carefully about your budget. South Africa is no longer cheap in Dollar terms. Your family will have to adjust to a completely different lifestyle and where you live and where you send your progny to school depends entirely on how much money you have.
You have literally zero chance of getting them into any of the top schools because there's been waiting lists for years.- But there is the American International school and the British one. There's a whole host of fairly good private schools- but forget about getting the top ones. SA residents get first dibs. Sorry.
That said there are about 4,000 Americans living in Cape Town either as residents, retirees, on work permits or as US consular employees. The US consular employees get the best selection of rental homes available in gated communities in the Southern Suburbs and if you are based in town you might want to consider the Atlantic Seaboard if you are very cash flush, or if you are not cash flush then, it's the Northern Suburbs or Southern suburbs for you. Best of luck in your decision. Leave your Paranoia at home. SA is a wonderful place to live and bring up children if you choose to make it so.
Africa is not for Sissies, but then again, Cape Town isn't really Africa...