Short version: massive bureaucratic inertia.

Long version: The Han Dynasty took power after a series of rebellions displaced the Legalist Qin Dynasty, and part of establishing a dynasty was picking an official state philosophy. Picking Confucianism made for a clean break with the Qin, and Emperor Gaozu of Han filled important positions with Confucians. These officials were responsible for recommending other officials for promotion, and usually picked candidates they agreed with; before long, the Confucians dominated the government, which shaped how those with aspirations of power were educated, how history was written, how art was made, and so on.

The real watershed moment came during the Sui Dynasty, where the Imperial Examinations were first implemented. Candidates for government jobs would be tested on their knowledge of the Confucian Classics, and their incredible difficulty arrogated enormous prestige to study of classical texts; later, during the Qing Dynasty, a province might allow ~4,000 man to take the exam, and perhaps 45 to pass it. This system lasted from the sixth century to the first decade of the twentieth; each successive generation of officials, teachers, authors, and artists was immersed in the philosophy from a very young age by aspirational parents. They even had a god of scholars, bureaucrats, and exams, called Wenchang Wang; these exams became deeply engrained in Han culture.

It's important to note here that Confucianism is an extremely broad term, and its practice in Imperial China wasn't necessarily straight from the Master himself. For one, every dynasty had its Code, the centerpiece of which was its list of punishments; Confucius's ideal was that the state and society be ordered through examples of morality, not on the basis of reward and punishment. During the Qing Dynasty, the Hanlin Academy Scholars who promulgated the official interpretation of the Classics revised the correct understanding quite often to suit the needs of the Dragon Throne

Essentially, Confucianism's popularity and flexibility made its study one of the best routes to success in the Sui Dynasty, which made it the best route during the Tang, then the Song, then the Ming, then the Qing.

The one of the best books on the subject of Confucianism on the examinations as an implement of the state is The Class of 1761 by Iona Man-Cheong, and What Remains by Tobie Meyer-Fong has a good chapter about the written word in Qing society.

Why did Plato hold on in the West?

Because someone can read Plato and come to 5000 different conclusions. And besides Karl Popper, most people like Plato. If you were an educated person in China, you were familiar with Confucius and he influenced you. If you were an educated person in Britain, you were familiar with Plato and he influenced you. Imagine if China studied Western thought, looked at how everyone drew from Aristotle and Plato, and then labeled everyone in the West a "Aristotlist-Platoist".

It'd sound absolutely ridiculous. But then again, there is no word in Chinese for "Confucianism".

I think the entire dominant narrative smacks of orientalism. Chinese/Japanese/Koreans are depicted as slavishly and uninventively worshiping one ancient philosophical figure while only Europeans are allowed to be depicted with intellectual agency.