Why did non-monarchic rule meet with so little success in ancient China?

I think it's because of their mentality with strong subordination to parents, boss, monarch, etc, who is senior by hierarchy. So, you can be either a "boss" or a "subordinate". If there is no monarch (the supreme "boss"), then there appear more than one equal persons, who have no "boss" above in the hierarchy, and this leads to chaos. When a chaos starts, there appears someone who is able to stop it and become a new supreme "boss" (monarch)

Although others may disagree, it seems that the success of "democracy" and "republican government" in places like Greece and Rome were do to special local factors that mostly did not exist in China.

Greece and ancient Rome were both hilly countries, which is to say poor for mass production of food using slave labor. With some other benefits of temperature and soil, they were good for producing delicate, high value-added crops like grapes (for wine) and olives (for oil) that are much more suitable for cultivation by "free" labor. In such a situation, one man was about as good as another, hence the idea of one (free) man, one vote.

The hills also made it harder for a central government to collect taxes, and easy for local-citizen armies to defeat much larger invading forces. Their relative wealth gave the peasants a strong incentive to enlist in such armies. Finally, the above mentioned goods were shipped around the Mediterranean by sea, far from land (and central government control).

Few of these conditions existed in China. The land was relatively flat, and subject to periodic invasions by northern hordes that local levies could not defeat. Only a strong central government (and army) could protect the people and engage in projects such as the Great Wall, that slowed down, but did not totally prevent such depredations. The one high value-added agricultural product, silk, was shipped by land, not be sea, and therefore did not foster "democratic" traditions that way that grapes and olives did.

Rome ceased being a Republic and became an Empire when a series of successful wars reduced foreign markets for wine and oil, and increased demand for (slave produced) grain from newly conquered territories such as Sicily. Essentially, Republican Rome fell when its best "clients" did.

It's probably not a coincidence that in Europe, the freest people lived in mountains (the Swiss), an island (England), or some peninsular combination of thew two (Rome and Greece). Likewise the freest parts of China were Tibet (until recently) and Taiwan, an island.

I think this is a question of wearing European/Western blinders.

For example, there were significant differences in governance between Qin (Brutal Legalism) and Han (Confucius) dynasties, and even more between the Han (Agrarian Economy) and Tang (Cosmopolitan, International Commercial Economy) dynasties. Tang and Yuan were different, as were Ming and Qing. The late Song can even be said to have been governed very similarly to the US today - wealthy business elites were very entwined with the political classes, and made many decisions about governance that were primarily interesting in self-enrichment of the elites, to the degradation of the lives of normal people. (I need to find the books I have read about these topics; I have completely forgot the names.)

Even more differences in governance methods were present during the 3 kingdoms period, and the bunch of dynasties that occurred in North China between the Tang and Yuan.

European political structure names don't exist for the differences, and European political theories don't readily adapt to well either. To seriously discuss the changes in Chinese governance methods for the last 2500 years, one would need to discard cultural biases and perform historical and political research from the ground up. To my knowledge, there is no single source where this occurs, nor any non-Chinese study that names the different governance changes.

It is a good idea to discard the commonly held notion that Chinese governance - and society at large - was unchanged for millennia.

It is also a good idea to investigate how a society actually functions rather than make comparative assumptions (A good book about this ). I have personally seen how responsive civic leaders in the PRC are to demands made by protesters a few years ago. It was a real eye opener. The people protested for a week, the "mayor" came out and asked what the protest was about, then helped them get what they wanted.

In the US, these people would have been tear gassed, beaten, and arrested. I witnessed the local leadership trying to find a solution to the problem, and changing their methods to suit the people. (While it certainly doesn't always end this way, but as an American it was nice to see the no tear gas or nightsticks deployed.) It is somewhat ironic that when I return to the US, people ask me what it's like to be in a non-democratic country.

I think you are not giving the Chinese their due credit. First, interestingly enough, there was a democratic (even anarchist) social movement that tried to create egalitarian communities in uncontrolled areas during the Warring States period; these were called the School of the Tillers:

In China around 400 B.C., for example, there was a philosophical movement that came to be known as the "School of the Tillers," which held that both merchants and government officials were both useless parasites, and attempted to create communities of equals where the only leadership would be by example, and the economy would be democratically regulated in unclaimed territories between the major states. Apparently, the movement was created by an alliance between renegade intellectuals who fled to such free villages and the peasant intellectuals they encountered there. Their ultimate aim appears to have been to gradually draw off defectors from surrounding kingdoms and thus, eventually, cause their collapse. This kind of encouragement of mass defection is a classic anarchist strategy. Needless to say they were not ultimately successful, but their ideas had enormous influence on court philosophers of later generations.

(David Graeber, The Democracy Project: A History, A Crisis, A Movement (New York: Spiegel & Grau, 2013), 188-189. Graeber cites Angus Graham, The Inner Chapters (Indianapolis: Hackett Publishing Co., 2001).)

Also, while not democratic, there was a strong history of peasant rebellions overthrowing the government. For instance, Graeber says (in the book Debt):

The two great threats to the authorities were always the same: the nomadic peoples to the north (who they systematically bribed, but who nonetheless periodically swept over and conquered sections of China) and popular unrest and rebellion. The latter was almost constant, and on a scale unknown anywhere else in human history. There were decades in Chinese history when the rate of recorded peasant uprisings was roughly 1.8 per hour [sic?]. What's more, such uprisings were frequently successful. Most of the most famous Chinese dynasties that were not the product of barbarian invasion (the Yuan or Qing) were originally peasant insurrections (the Han, Tang, Sung, and Ming). In no other part of the world do we see anything like this. As a result, Chinese statecraft ultimately came down to funneling enough resources to the cities to feed the urban population and keep the nomads at bay, without causing a notoriously contumacious rural population to rise up in arms. The official Confucian ideology of patriarchal authority, equal opportunity, promotion of agriculture, light taxes, and careful government control of merchants seemed expressly designed to appeal to the interests and sensibilities of a (potentially rebellions) rural patriarch.

I think the answer lies in the Chinese idea of the Mandate of Heaven. Literally the ruler was allowed to rule because heaven blessed him. This idea goes all the way back to the Zhou dynasty (1046–256 BC). Added to this was the philosophy of Confucian Relationships. This gave a very fixed notion of who had what position in society and why. The King or Emperor had an honour but also a duty. Those under him likewise. This was looked on as an ideal of perfection and harmony. When today, Hu Jintao talks about creating a harmonious society, he is harking back all the way to Confucius. These ideas, developed early in Chinese history, have long be a powerful force of conservative rule.

Towards the end of the Warring States Period, a system of Legalism became popular. This was especially promoted by the Qin Emperor and on into the Han Dynasty. Legalism still has its influence in today politics. The Qin tried to suppress Confucianism in favour of Legalism. Legalism also gives a strong distinct role to the ruler of the state. However the rule of law is emphasized. During the Qin, the aristocracy was abolished and a system of meritocracy introduced for all ranks below the Emperor. This change in governmental system as significant in seeing the Qin become the empire they became.

As we advance to the Tang dynasty, Buddhism arrived in China and this brought with it yet more philosophical ideas about the nature of government. It remained a monarchy at the highest levels but the system of governance was heavily influenced by Buddhist teachings.

In the 19th century, we see the Taiping Rebellion (1850 to 1864) in southern and central China. Latterly based around Nanjing. While in practice this was a monarchy or military dictatorship might be a better phrase, it was in theory to be a new form of government. They were based around a corrupted form of Christianity. They also had ideas of shared property and land redistribution. However, the never ending war and the short life of the so called "Heavenly Kingdom" meant that none of these ideas ever got properly implemented.

At the tail end of the 19th and beginning of the 20th Centuries, we see a weakening Qing Empire attempting to reform. There were attempts to bring in a constitutional monarchy with a parliament, similar to some western governments. However, politic plays within the Qing fell in favour of the conservative elements and nothing substantive was done. By 1911 there was revolution and in 1912 the emperor abdicated. Over the next 50 years, Chinese seems to have tried out every possible kind of government imaginable. Most didn't last long and we all know what ideology won in the end.

So although China has always had an emperor or monarchy, to think of China as having a single system of government for 2000 years is incorrect. The system under which that monarchy operates and the form of government at lower level has often changed and been reformed.