but that is already 1 too many for a flat earth

You can do this same sort of thing in the Northern Hemisphere. It just gets more obvious the further you go south.

For example, check a direct flight from LA to Seoul. On a Gleason map (the most common flat earth map, though there are a few others), flights between those two should be going well over Alaska and parts of Russia. On a great circle route, they barely go over the Aleutians and don't go into Russian airspace at all. Guess which one flights actually use?

Is there one that isn’t nonstop (excluding stopovers in New Zealand- not because it doesn’t count, but because I’m wondering whether there’s a place to stop between South America and Oceania)?