Lost South African passport in the USA and emergency travel

The VFS Global service list does not include an Emergency Travel Certificate [source]. The certificate is handled by the Embassy. It is a one-way travel document to South Africa only, valid for a maximum of three months from issue [source].

• Completed DHA-73 and DHA-9 (a copy of the original DHA-9 can be used; prints must be clear)
• 2 photos (colour, no smile, mouth closed, no glasses)
• Copy of passport, South African birth certificate or identity document
• Proof of travel arrangements and a written reason for urgent travel to South Africa
• Fee $12 money order payable to the South African Embassy, plus a prepaid envelope for collection
• The Embassy form says the minimum processing period is 7 business days. Submit or mail the application to the South African Embassy, 3051 Massachusetts Avenue NW, Washington, DC 20008.

[source]

The Department of Home Affairs says an Emergency Travel Certificate issued abroad usually takes 3 to 5 working days [source].

The New York and Los Angeles immigration pages use the same sentence. They say the embassy is open for this certificate Monday to Friday, 9:00 a.m. to 12:00 p.m. and 1:30 p.m. to 3:30 p.m., without an appointment, and that the process takes 2 to 5 days. They say "the embassy". Those hours are not the consular hours on the New York or Los Angeles home pages, so this guide does not use 2 to 5 days as the timing at those consulates [source, source].

Home Affairs will not approve an Emergency Travel Certificate for applicants eligible for a first identity document (turning 16 and above) who have not applied for it [source].

DHA says a South African living abroad who still holds South African citizenship needs a valid South African passport to travel to South Africa, and that proof of a passport application is not accepted for travel. See citizenship [source].