Please note that you can currently fly from South America to Asia (not going to North America) with several carriers. Some options I list below:

AR will fly you from EZE to Auckland - you can then go onto Asia proper.

MH will fly you from EZE to KL via South America.

SA will fly you from EZE to South Africa, you can then take SQ or a ton of other carriers to Asia.

RG will fly your from EZE and Brazil to Asia, transpac or transatlantic, no US connection needed.

Qantas was also doing EZE-Sydney flights, I dont know if these have been discontinued though.....

Im sure there are a couple of more options that Im not listing here.... and dont forget the european connection possibilities...