Both aztecs and their rivals were decimated by disease, that was the key.

In the long-run, in terms of why there wasn't a resurgence of native polities, sure. In the short-term, the ability of a small expeditionary force of Europeans to exercise disproportionate force (making them a desirable ally for the Aztecs' enemies) was the key element of the overthrow of the Aztec polity, not disease.

Given time, they would have been able to master horse riding, armor and more modern weapons, as they actually did in one of the last battles in Xochimilco just a year after the fall of Tenochtitlan, but too few remained to make a difference.

I'm unfamiliar with that particular battle, but I'm familiar with later examples. Those weapons also need to be produced in large enough quantities to make a difference, as well as proficiency with them developed. Horse riding is a notoriously difficult skill to master, and metallurgy itself is far from simple. Being able to use captured equipment at a basic level is not nearly the same as being proficient with it, or being able to equip an entire cohesive military unit with it. Disease does not explain battlefield imbalances, which is what I'm referencing.