If you're having issues, look up professional US account opening service providers. Costs start at 2K for account opening assistance. Plenty of choice there, especially in Delaware and Florida. There's absolutely no argument to think or suggest that it's not possible.
Second, why would you want to open an account with a U.S. address as a South African resident? If you're trying to hide from South African gov. with a U.S. address, it's doable, but you'll get in trouble with the IRS if you make the bank believe you're a U.S. person. You outlined what you want to do, but your ultimate goals are still unclear.
If you're not hiding anything, your optimal choice is to open a U.S. account with your South African address.
Due to low interest rates (which now hit the U.S.) quality high street banks will become a little pickier, similar to how they've been Singapore and HK over the last 3 years. For high street banking as a non-U.S. person, look to deposit 100K+ USD, use a bit of FX at their crappy rates, and you'll get your personal account. For anything less, trashy options like TransferWise and Payoneer are available.