They had lots of exposed areas, and those obsidian swords of the Aztecs are devastating against exposed flesh.
There was no exposed flesh. Neither the aztec warriors nor the Conquistadors were running around fighting naked. The conquistadors had padded, leather, and metal armor and the Aztecs had their cloth armor which, while effective against obsidian weapons, did virtually nothing against crossbows, spanish swords, musket balls, etc. (Of course, hey, neither did plate armor tbh).

There is little doubt that the armor and weapons of the conquistadors was vastly superior to that of the Aztecs. There is almost no way they could have won, otherwise, and had the Aztec weaponry been superior, you can bet the Spaniards would have armed themselves with it after their first battle. The Aztec troops and weapons were nearly wholly ineffective against the conquistadors, for a variety of reasons.

CA thought this would make poor gameplay, so the changed the Aztecs in MTW to be a lot more powerful and more challenging than they were in real life. The game is not really a 'historical simulation' but the new world is one of the least historical parts about it. Just accept that it's a fun little side show with a 'new world flavor' :)

If you're so obsessed with the efficacy of Spanish plate armor, consider that the Aztecs had a weapon called the atlatl - mostly just a stick that simulated an extra length of arm and so could add leverage to a thrown spear. Supposedly they were so powerful that they could send a dart not only through armor, but all the way through a man. Consider that it wasn't a terribly complicated weapon to use, thus you could arm thousands of commoners with it - much like a crossbow. That would give any conquistador pause ;)
If the atlatl had actually been that good Cortez entire tiny band would have been annhilated in the first volley of missiles from the Aztecs. However, we read that the Aztec missile volleys (of which the vast majority of their 'soldiers' were equipped for) were surprisingly ineffective.

Regarding the numbers issue: Yes, Cortez did have a few hundred men vs many thousand, most of the time. However his men were all superbly trained armed and armored, and the vast bulk of the enemy forces were extremely lightly armed and were not the 'warrior class' of their society that ran around in the armor and obsidian clubs, etc. That number was much smaller. They didn't do terribly well vs the Spaniards either, but once *they* were defeated the lesser Aztecs pretty much decided things were unwinnable.