2. The scholar-gentry not doing away with the emperor but basically turning him into a figurehead. This has probably happened de facto in the past, whether by other members of the imperial family, or by the scholars, or by the eunuchs.
This has happened many times actually, Dong Zhou, Cao Cao, Sima Zhao, Liu Yu, and many others for starters from military figures and politicians that were able to use the power of their military and their political guile in order to take command of the Empire, puppetizing the Emperor. With Dong Zhou there was a completely lack of guile leading to a number of revolts against him, but with Cao Cao he was able to take command of the Empire or at least half of it and rule it until his death. He climbed the ranks slowly, becoming a Prince, but never became Emperor. His son did so almost immediately after his father's death. Sima Zhao was an established nobleman, the second son of Sima Yi, and managed to do the same as Cao Cao, though with less reform and more politics and warfare. Liu Yu was a military commander that gained prominence in the Jin Dynasty, that Sima Zhao's son would later found, and would grasp the government and control it for several years before making himself Emperor. This is just the space of a couple hundred years in order to paint the picture that such actions were common, particularly in eras of fragmentation.
Instances where family controlled and manipulated family were also common, Wu Zetian the only female Emperor of China started out through manipulation of her own children and step children.
The Eunuchs almost were known to manipulate Emperors for generations, but their control always waxed and waned. Its hard to make them permanent authorities inside of China because most of their power comes from manipulating not just the Emperor, but the Emperor's family. When they become to overt they normally end up dead. Further, the position of Eunuch is at its nature domestic, without martial command, while they were able to manipulate military officials through bribery and appointments, they never were able to directly control it. They also were almost never really able to leave the capital without extraordinary circumstance, nor did they ever really manage to attain authority and official appointment outside of the capital. This leaves them in a very vulnerable position should anyone with an army decide to be rid of them.
The scholars, however, did not do this. Being a scholar alone was never enough to be able to grasp the reigns of government and manipulate it behind the scenes or even overtly because of the Emperor's family and political rivals. In order to fully manipulate the Emperor and his family without much complaint is to win the manipulation and power in military campaigns or acrewing a very long career with a reputation for loyalty and talent. Otherwise the only way for a scholar to gain control would be to be directly related to the Emperor or another very powerful minister. Many noted scholars did manipulate Emperors and even the Empire, but this was only in addition to strong military or political influence.
Basically, I know emperors have their place, to serve as the Sons of Heaven and to start dynasties and to lead military campaigns and do all of the things that a despot does. But seeing as how China was usually administered by scholar-bureaucrats, why couldn't those guys rule instead?
They often did, with the consensus of the government by manipulation or by the Emperor's own apathy. Several noted Emperors in the Tang, Han, Song, and other dynasties simply delegated all internal matters to the actual beurocracy. This was actually a common practice. The Emperor was the head of government, but like all governments the head of government does not manage everything themselves. The Confucian bureaucracy operated under the Emperor's direct authority as well as in the Emperor's absence and recluse. Its not quite "in-command", de jure, but its not without authority. However it was antethetical to Confucian Ideals [Tianxia/Five Relationships/Mandate of Heaven] to remove an Emperor and replace it with something else, and remained so until well into the European Age of colonialism. Even then the ideas didn't die until the cultural revolution and the PRC as well as the ROC taking command preceding that.
Though I guess this is like saying "in medieval Europe the Catholic Church did much of the administration in the Holy Roman Empire, why don't the monks and bishops get rid of the kings?" But the difference there is that in post-Zhou China the class structure wasn't based on military strength like feudalism was.
Yes and No, while legitimacy and authority came without the need of an army, the army was almost always used as a tool to gain control of the government. Often times the regents that would manipulate the Emperor at the behest of the Dowager were powerful leaders in the military. In times of fragmentation Empires were forged from the sword more so than the Bureaucracy with the Confucians attempting to legitimize actions already taken place rather than guiding said actions.