I might be wrong on this, but this is my opinion on the "disease" reason:

If the Aztecs and Incas were conquered because of disease, how were the Aztecs and Incas still able to send so many reinforcements? The Aztecs and Incas always had more than enough soldiers available, even when disease was rampant. During the siege of Tenotitlian, for example, the Spanish and their allies were severely outnumbered, yet the city fell to them.