New Zealand and Tijuana, Mexico
How can I find flights that exclude stops in the USA?
Let's start getting from Australia/New Zealand to the continent. Google Flights can search non stop Auckland - South America flights and shows the only flight existing is into Santiago, quite a way from Mexico. It can also search for Auckland - Central America but that comes back empty. But alas, this is the only way if you want to avoid North America. If you move the map around, Vancouver also shows up:
Via Santiago it's 10 095 miles, via Vancouver, it's 9492 miles. That's not a significant difference -- but both add a sizable distance to the 6800 miles between Auckland and Mexico City but what can one do? You can test with the Great Circle Mapper that any major hub elsewhere is even worse: Tokyo is over 12000 miles, the rest like Hong Kong, Singapore, Dubai is over 15 000 miles.
That established, while Wikipedia in general is not a reliable source to say the least, airports are neutral enough that not much trolling is going on so it's a reasonable good resource for this sort of research. We can look at the Tijuana airport article and find this curious tidbit:
International service ended in early 2020 (due to COVID-19 pandemic) after Hainan announced the cessation of all flights to Mexico.
So we will need to transfer within Mexico, no matter what. For connection, Mexico City is the evident choice, it is an extremely large airport -- the 16th busiest in the world.
You can try searching for flights transferring in Vancouver or Santiago
first and in Mexico City second. Or you can even make the second connection point "anything" in hopes it'll find some other point than Mexico City (it doesn't):
Or you can buy two tickets. When flying on an unprotected connection, especially with flights this long and expensive always presume you will be bumped to the next flight and buy the connecting ticket correspondingly. Spend a night in the connecting city, in other words.
So fly to Vancouver or Santiago, sleep there and then fly to Tijuana via Mexico City. I would prefer this even if I had a single ticket because the first leg is just brutal and a shower and a bed is very, very enticing after such a long flight.
Also, as a small footnote, Vancouver is one of the most beautiful cities in the world but I might be slightly biased in this.