Tbf is only 50k men cause last time the Chinese invade Korea they lost hundreds of thousand due to logistical issue (sui dynasty) so 50k make sense as is just enough supply issues isn’t that bad and big enough to stop the Japanese from invading further. Perfect for an intervention force.
Also quite annoyed people are treating China as a paper tiger
The Great Ming, did not make that decision due to Sui past precedence but because the Great Ming had difficulties actually raising an army. Finances were poor, factions in the Great Ming urged caution or blamed the Koreans for the war and hence did not care, common view by the Ming was that passive wu-wei policy in diplomacy was ideal and finally the Great Ming armies were very atrophied and declined except in choice areas. Great Ming was definitely and indeed a paper tiger in terms of needing to combat enemies in warfare. They may be able to get serious and crush an enemy in a long protracted war, but that may require a serious realignment in the Central Court that otherwise is disinterested in the external affairs outside of the capitol and the idealisms of Confucianism.
Mexico and China are so differents that this is like you comparing a kitten with a tiger.
"Mexico" did not have a navy, China does.
Mexico was much less populated than China, and this was only exacerbated by epidemics, which is not going to happen in China.
Mexico suffered a technological gap that China does not have (in fact, they are probably more advanced than the Spanish in various technological fields).
"Mexico" was a chaos of proto-states fed up with their ruler and waiting for a spark to start mass rebellions, as well as willing to ally with anyone against the hated Aztecs.
China is an state, and their people, even if could hated its ruler, would hate invading foreigners even more, and believed in the eventual unity of all of China.
So there will be no one agreeing with the Spaniards to make way for them.
So no, this it's not comparable at all.
The Mexica Triple Alliance was defeated by a very fearsome Spanish army and suffered a catastrophic and potentially avoidable battle at Otumba, however I think it should be noted that the Mexica Triple Alliance was a more effective martial state than the Great Ming. Despite the definite situation where there were certain independent states in Mexico to oppose the Mexica Triple Alliance, the Mexica could quickly rally several armies of large size and high skill to crush enemies and further Mexica state apparatus, keenly focused on martial expansionism, espionage and political maneuvers, was not caught aback by Spanish intervention. Great Ming on the other hand would likely have difficulties understanding the threat that opposed them until it was a dire issue and this was the same problem faced by the Great Qing in their later phase in the middle 1800s. Very nearly, the Great Ming were almost caught unaware by the Japanese who not only advanced through a close tributary but also were closing in on the border only a short distance from the capitol. A Spanish invasion of the southern coast of the Great Ming might barely warrant a Great Ming response for months and said response will be haphazard at best and surrounded by passive negotiation.
As for the use of dissent local elements, the Jurchen/Manchu spent the better part of 2 decades initially and then another 2 decades of intermarriage, bribery, and espionage to attract Han defectors. This did not require the defectors to change their religion to stop culturally significant practices like ancestor worship (which tanked Catholicism's chances in China OTL) or trust random foreigners that they hadn't had long term relations with and knew would give them ample returns for their loyalty. Whereas the Aztecs were actively hated by their subjects and both the Aztecs and Inca were in active collapse due to pandemics during the Spanish invasion and vastly outgunned, forcing locals to choose between the Spaniards or chaos that would inevitably end in subjugation, the Ming were still stable enough to lose their capital to rebels, then northern barbarians, and still survive and fight on for another 4 decades.
The Great Ming survived in the south as a form of local warlords who upheld a faux Great Ming. The Central Court was destroyed, and in theory the Central Court in this hypothetical scenario would probably allow local officials in southern China to deal with a Spanish attack and only take action if there were inclement issues and otherwise not feel much danger. Regarding the Mexica, the other states may have disliked the Mexica, but no particular Indigenous state in Mexico was willing to combat the Mexica alongside Cortes in fact until Cortes managed a phenomenal victory at Otumba. Mexica had no issues with epidemics until the final stages of the war when the result was already decided due to Mexica defeat at Otumba.
For Mexico and the Andes i've seen it mentioned more than once that the deciding factor was not firearms but swords and armor made out of steel. There's no real technological gap in that.
There might be though. Spanish forces will not face a standard Great Ming army most likely, but a sort of Yong-Ying militia force, which will not carry heavy armor of any kind and will use primarily cold weapons of relatively low quality. While the Great Ming possessed supposed large numbers of trained soldiers, the reality was that most of their soldiers were relative serfs who possessed poor training and had squalid command and tended to spend most of their year doing odd jobs to make up for the fact that Great Ming budget was notoriously frugal.
You mean the Chinese who produced their own firearms of their own design? The Chinese Breech Loading Canons were made independently of the Europeans with normal trading of ideas through the Silk Road with the Turkish merchants buying the ideas from the Portuguese who sold it to the Iranians who traded it with the Chinese in the 1520s in return for silk luxuries. That's called normal commerce. It's also how Chinese gunpowder reached Europe from East Asia in the first place. Or Ming matchlocks which outmatched the Dutch and French ones which they made up on their own in the 1540s under the tutelage of Qi Jiguang?
Or the Japanese? Who since the 1270s independently traded firearms with China and Korea that they had their own stocks? It was simply not used to the same scale as Europe because the topography of Japan lent itself until the 1480s for more slash and flee attacks that were so common in the Ashikaga Shogunate?
Or the Koreans, who invented the Byeol-hwangja-chongtong which when used by the Chinese devastated the Dutch in Taiwan even 100 years later? And the fact that they had entire caches of firearms that as per Science and Technology in Korean History: Excursions, Innovations, and Issues by Song-Nae Pak (2005) matched the European ones used by the Wokou pirates and Japanese with every punch? To say nothing of the Hwacha itself.
Or the Iranians under the Safavids, one of the legendary Gunpowder Empires, which stood toe to toe with the Ottomans who themselves stood toe to toe with their European rivals? The Safavids definitely did use captured Ottoman firearms to supplement and improve their own, but they also had thousands of foundries of their own which made Safavid matchlocks and canons, most famously the ترکیدن شیر (Sher) that pushed the Portuguese privateers out?
Or the Delhi Sultanate or Suri Empire or finally the Mughals? The very same Mughals which in 1656 produced the most artillery and matchlocks in the entire world, even outstripping half of Europe? (Source: Mughal Warfare: Indian Frontiers and High Roads to Empire, 1500–1700 by Jos Gommans)
The tradition of Asian great powers importing firearms and materials for warfare from Europeans is 90% - 95% a post-1700 phenomenon. Previous to that the Asian Great Powers - the Mughals, Chinese, Japanese, Iranians and Ottomans made their own, supplied their own, designed their own firearms which were more than a match for European firearms. The Great Divergence is dated at 1700 AD for a very good reason. The Collapse of the Ming Dynasty, the Start of the Edo Period and the Decline of the Mughal Empire collectively alongside the Asfarid Transition in Iran had much more to do with the technological gap than anything the Europeans had done previously or at all. (Source for Great Divergence: Great Divergence and Great Convergence: A Global Perspective by Leonid Grinin
We should compare these empires though not by what they produced but how their imperial power existed at the time. Simply because the Great Ming had skilled foundries, does not meant hat these productions were correctly utilized. Further, simply because these realms produced canons and ones that could be useful, does not mean that their canons are equal to that which Spain could bring. Likewise, Ming doctrine in the period is what is the problem, as if the Spanish have an in depth plan and coordinate their attacks well, they could tack on a huge victory and cause a cascading event in the Great Ming of rebellions and civil unrest that could see the dynasty overthrown.
So? The Jixiao Xinshu, or the wartime guide to all Ming Soldiers in the 1550s and 1560s issued the Volley Fire of the Star Prong doctrine that ensured 1080 musketeers firing volley fires repeatedly at enemy positions, with at least 600 musketeers firing simultaneously while the rest reloaded. These musketeers were hidden in infantry divisions of 3400 where the rest supported the musketeers whilst the cavalry conducted a two-pronged assault from the side with artillery canon fire from the back. Devastated the Japanese at Pyongyang to the horror of the Portuguese and Dutch onlookers. Or the Japanese Five Columns which arrayed the Japanese formations in bent lines that increased their range of muskets from 80m to around 140m which laid waste to the unsuspected Koreans during the early stages of the war? Both of these were tried to be replicated in Europe in the early 1600s and was only replicated successfully by Gustavus Adolphus partially during the 30 years war and then fully by the Habsburgs and Swedes during the 1680s. This is fact. (Source - The Gunpowder Age: China, Military Innovation, and the Rise of the West in World History by Tonio Andrade, and Firearms: A Global History to 1700 by Kenneth Chase and Jixiao Xinshu
An elite Great Ming army was not insufficiently skilled or trained to defeat a Spanish incursion, but we should understand the vast relative distance... Assuming the Spanish invade the far south, the Great Ming would be over 2,000 km. In Korea, the Great Ming had a distance of 800 km and their elite armies nearby and the heartland of their powerbase to draw from. Great Ming finally began to take interest in Japanese expansionism when Japanese forces were present just 700 km from Beijing, and with an army larger than what the Great Ming had stationed in Beijing. I seriously doubt that the Ming will take notice of the problem until the Spanish make considerable gains against militia in the south and then by that time, it may cause civil issues for the Great Ming and destabilize the empire regardless of the decision that they make.
Otherwise, I would put the success of the Spanish in such an invasion to be minimal. While the Great Ming may be a relative paper tiger in terms of its ability to react to threats near it, the Spanish cannot afford the planning and investment needed to make a good try at invasion. Invasion would require a very precise set of plans and skill that the Spanish may not be able to tolerate. Planning in terms of building alliances and using mercantile connections to build alliances with secret societies, bandits, disaffected officials in the region and pirates. Further, a coordination with the Japanese would highly augment their chances but such attempts would be blocked by the Portuguese.