I'm afraid that the answer is, at this time, clearly, resolutely, and absolutely, no.
There are no exceptions for transit to the current travel bans, whatever the duration of transit.
There is no airside transit in any airport in the US. All passengers, whatever their final destination, will go through immigration (passport/visa control), reclaim their luggage, if any, and go through customs, before they can move on. In the vast majority of cases (and that includes LAX), passengers will end up landside in the public area, before they go to departures and then go through security to go back airside.
Even if the two flights were on the same ticket, you would not be allowed to board a flight to the US if you have been in any of the banned countries in the last 14 days and you are not an exempt person (US citizen or permanent resident and a few other similar exceptions).
Even if there was an exception for transit (which doesn’t exist), with flights on separate tickets, the first airline would consider your final destination to be the US, and again, you would not be allowed to board.
There is talk about lifting restrictions for some countries, but I have no idea if South Africa is supposed to be part of those changes, there is still no date ("early November" is all we have)ù, and no details.