I'm... not sure I'm convinced. Why would you breed a small, yapping dog for guard duty when you could breed a large, barking dog that can also put up a fight if it needs to?
I've tried to post this 3 times, but my phone ate it all 3 times.
Tl;Dr:
Chihuahuas are named after a Mexican State that is named after a desert. Big dogs don't do well in deserts.
Sure, you're not going to be very successful breeding for a Great Dane in northern Mexico. But jackals weigh 20 pounds, coyotes are 25 or so, dingoes are 30-40, and African wild dogs weigh up to 50. All of those do just fine in the desert--so clearly chihuahuas are nowhere near the ceiling the ecosystem can support.