'Caliphate Empire'
call it the Caliphate only or (dynasty) Empire/Caliphate (e.g. Umayyad Empire), please, it hurts my OCD.
A better POD would've been the Black Death sweeping all of Europe first in the mid-14th century, which would destroy Europe's population. Then you have to make sure that the Moroccans and Andalusians don't get interested in the New World and keep China, Korea and Japan isolated. or like Sunset Invasion (in CK2), Vikings enter North America, allowing for the Mesoamericans to advance, although that might be a little too weird, allowing them to acquire technology from the Norsemen and eventually turn on them
Kim Robinson Stanley's Years of Rice and Salt is one read of fiction in what would happen if Europe had 99% of its population got wiped out by the Black Death. But this speculates that the Sino-Muslim world should be able to inevitably colonise the New World eventually, which would defeat this purpose of modernising the Aztec Empire. There's probably no way around it without Muslims or the Chinese helping the Aztec Empire to forcibly modernise like Japan. Then the Aztec Empire could eventually become like Meiji Japan. (this comment was removed by YouTube for some reason as spam, prompting me to post this instead)
Thus the third option seems more of a possibility than the previous two options, though it is the first option that this third option is derived from, with the black death killing 99%.
I do have an alternative POD but this requires the Aztecs (at least the elite) to at least be sympathetic to the Spaniards and are more resistant to illnesses from the Old World (which will be ASB, aka implausible but uh I don't care). So Hernan Cortes would make the Aztecs a puppet state essentially, making him a 'god' under the Aztec king's interpretation and the current Aztec king isn't stoned to death by his opposition. The Aztecs would have to avoid being annexed by Spain into its New Spain Viceroyalty. They would acquire firearms and modernise to fight back against the Viceroyalty, allowing them to form their own kingdom and eventually get other colonial nations to ally the Aztecs.
Essentially they'll be like Japan/Thailand/Ethiopia and would become heavily modernised, butterflying Mexico out as a state. It could've been Madagascar had the Malagasy leaders before being subjugated by the French government entirely weren't so complicit. Ethiopia only survived its fate because Italians were defeated in the Battle of Adwa; this could be said same to the Aztecs.
its extremely oversimplified but I think its possible to pull modernisation of the Aztecs earlier. I did try to meddle with medieval European politics briefly within 200 years without having to skip centuries of time. its more of learning politics by following events depicted in Wikipedia and eventually straying.
This could be applied to a 'Modernisation of Industrial Europe' on 1789. I've been thinking of doing similar scenarios that are closer to today but rather than wait for 200 years of industrialisation, it'd be 100 years until we reach 2020 levels.