Pucker factor meter was no doubt pegged for the Spaniards
In the end. Cortez’ invasion, unwittingly, brought a far worse fate. For what ultimately defeated the Aztecs (and allowed Cortez to finally capture Tenochtitlan), was the plague of European diseases that the Spaniards unknowingly brought. And it spared no one among his allies. It is estimated that, in the end, 90% of the American Indian population of the New World (NINETY - you read that right) was wiped out by these diseases.
This is why, incidentally, when the English colonists arrived in numbers on the East Coast, they found comparatively few Indian tribes where stories of a few earlier expeditions talked of thriving cultures - the plagues brought by those initial explorers had, within a couple of decades, wiped them out.
That’s textbook tactics though. If you can’t overcome a superior force, ally with the minority faction and fight. The “stealing the flag” thing sounds a bit confabulated though.