Do they see it as the gods punishing the Aztecs for their wicked ways and granting the Chinese dominion over them? Or do the Chinese think it is simple coincidence (hard to believe that they would). The most likely out come would be divine providence, considering that spirits were believed to be in everything and thus everything was the work of spirits.
Poor phrasing. The epidemic might be considered as punishment from
Heaven for the Aztec
ruler's personal transgressions, though that would really not mean much politically since, once again, the Chinese never did care all that much about the events in far-off countries, and would not really
want to take over far off Mexico. I really don't see them regarding this as an omen to seize Aztec lands, because to my knowledge they never applied the Mandate of Heaven doctrine with a view towards expansion outside of the traditional boundaries of China. What's rather more plausible is that they might advise some of the local malcontents to seize power and receive the title of King of Mexico from the Chinese Emperor himself - thus becoming truly legitimate rulers and saving their country from all manner of troubles as long as they behave. An intriguing alternative, especially good for the exiled courtier scenario, is that the leader of the expedition might simply try and seize power there himself - I don't see why he couldn't if Cortez could pull it off, though holding on might be more of an issue.
Then, ofcourse, there is the fact that Chinese medicine was quite advanced and, in fact, advancing quite significantly in the early Ming period (that was when the first inoculations against smallpox were made, IRC, at least in China). Scholars might very well figure out the real explanation, sort of. Probably won't change much, though.
All in all, the Chinese
Emperor would want obeisance and a ritual tribute. Maybe the eunuchs might want to get something more, but then again, gold was not as valued in China as it was in Europe, so I'm not sure if they would be tempted to organise any serious attack. I agree that the first contact hinges entirely on the expedition leader. Lastly, in the long-term America might become a macro-Taiwan - a refuge for pirates and malcontents fleeing from Chinese authorities, provided they have ships that can go there. That alone should have some interesting consequences for the region, especially if the Europeans arrive more or less on schedule - imagine clashing with Chinese mercenaries - or, hell, Chinese
rulers - in Mexico, or having to cope with wako pirates at Panama.
Government-driven Chinese colonisation, as opposed to the independent or even anti-government diaspora, will probably be doomed, though; I don't think they would make a bigger effort than they did in the Indian Ocean, but even if they
tried to resettle people there, such colonies would probably not be cost-effective and would eventually be abandoned altogether or just to their own devices.
To sum up, the trans-oceanic Chinese Empire is almost definitely impossible, at least without much earlier changes. However, a new China separate from the old China might very well come into existence, which IMHO might be more interesting than the original idea (especially if you make the Ming get overstretched and collapse earlier). Europeans will still arrive, but will find the New World rather more difficult to conquer, because of a smaller technological advantage, a lesser surprise effect, an already recovering population and, last but not least, American states with more experience in dealing with other powers - it has often been pointed out that this lack of "diplomatic experience" was a major factor in the destruction of the Incan Empire, in particular. So we might have surviving Mexican and Andean civilisations (if maybe in somewhat altered forms), an
ethnically Chinese (and Japanese?) West Coast of North America, and Europeans in the eastern parts of North and South America and the Caribbean.
The "technological inferiority" is not nearly as wide a gap as you make it out to be.
Out of curiosity, just how wide does he make it out to be? He simply mentioned that it's there and is significant - surely you won't deny that?