As you've discovered only 2 flights (both less than daily) between South America and SWP. Thus can be difficult to get, but far from impossible.

Assuming you are using the online tool, try EZE-SYD, SCL-SYD (goes via AKL) and SCL-AKL separately. On the LA flight availability can differ between SCL-SYD and SCL-AKL. From AKL you can either take another flight to SYD/MEL/BNE & then on to Asia or fly CX to HKG directly.