Cortez and his 300-odd men could never have taken down the Aztecs, were it not for how all their neighbors hated them so passionately that they'd rather be ruled by smelly, strange-looking, gold-hungry space aliens.
The diseases certainly had an effect (after the Night of Sorrow and before the conquest of Tenochtitlan, smallpox struck the whole Valley of Mexico); but Tlaxcala had an effect too, and if the Aztecs' subject peoples had rallied to their arms in the same way the Tlaxcalans rallied to the Spanish (after some initial fighting against them), there's no way Cortez could have won.