I just checked the episode. Phil said travellers cannot book International flights from the Ushuaia Airport.
I don't care what Phil said. The key is whether you can book any international flights from Ushuaia. The answer is that you can seasonally (that means not during the winter in Patagonia when few travelers want to go there). Here are the airlines and desintations served by Ushuaia:
Aerolíneas Argentinas -Buenos Aires-Aeroparque, Buenos Aires-Ezeiza, El Calafate, Rio Gallegos, Trelew
LADE -Comodoro Rivadavia. El Calafate, Puerto Deseado, Puerto San Julián, Puerto Santa Cruz, Rio Gallegos, Rio Grande
LAN Airlines -Seasonal: Punta Arenas
LAN Argentina- Buenos Aires-Aeroparque, El Calafate
All of the above are domestic within Argentina except Punta Arenas, Chile.
Now, this does not answer the real intent of the original question, which is whether you can book through to southern Africa from Ushuaia. You can book to either Buenos Aires or Brazil and from there to Maputo. The list of connection destinations possible starting from Ushuaia is:
Australia LAN Airlines, Aerolineas Argentinas
Paraguay Aerolineas Argentinas
Peru LAN Airlines, Aerolineas Argentinas
Spain Aerolineas Argentinas
United States LAN Airlines, Aerolineas Argentinas
Uruguay Aerolineas Argentinas
Mexico Aerolineas Argentinas
Italy Aerolineas Argentinas
Bolivia Aerolineas Argentinas
Brazil TAM Linhas Aeras, LAN Airlines, Aerolineas Argentinas
Chile LAN Airlines, Aerolineas Argentinas
Colombia Aerolineas Argentinas
Ecuador LAN Airlines
Venezuela Aerolineas Argentinas