The difference you have pointed out here is not really one between China and "the west," since some countries that are usually considered "western" do, in fact, require Classical Greek and/or Latin as mandatory subjects in schools. Notably, students in Greece today are usually required to study Attic Greek, and students in Italy are usually required to study Classical Latin. The difference here, then, is really one between countries like China, Greece, and Italy and Anglophone countries like the United States, Canada, the United Kingdom, and Australia.

Furthermore, for most of "western" history, Greek and Latin formed the entire basis for both a grammar school and a university education. In the seventeenth and eighteenth centuries, the primary purpose of a formal education in the English-speaking world was to prove that someone was a true gentleman—what the ancient Greeks would call a καλὸς κἀγαθός—and knowing Greek and Latin was absolutely essential to the early modern conception of what an educated gentleman was supposed to be.

As such, when Harvard University was founded in 1636, anyone applying for undergraduate admission was required to already speak fluent Latin and be able to write perfectly all the paradigms of Classical Greek. If a person wasn't already fluent in Latin, then they were considered obviously ineligible for admission. A document from 1642 describes the qualifications for admission at Harvard as follows (as quoted in this 1914 article in The Harvard Crimson):

"When any Schollar is able to Read Tully [i.e., Cicero] or such like classical Latine Authour ex tempore, & make and speake true Latin in verse and prose suo (ut aiunt) Marte, and decline perfectly the paradigmes of Nounes and verbes in ye Greeke toungue then may hee bee admitted into ye Colledge, nor shall any claime admission before such qualifications."

Other early modern universities had similar language requirements for all students seeking admission.

Over the course of the late eighteenth, nineteenth, and twentieth centuries, though, the way that people in the English-speaking world thought about education changed. As the point of a formal education became less and less about aspiring to be a gentleman and more about acquiring "practical" knowledge that would be useful in one's life and career, people began to argue that forcing students to learn dead languages takes an enormous among of time, that knowing such languages has little-to-no "practical" value, and that it would make the best use of students' time to cut the requirement of Latin and Greek and instead teach them subjects that have modern "practical" applications.

The American writer Thomas Paine (lived 1737 – 1809) presents a relatively early version of this argument in his book The Age of Reason; Being an Investigation of True and Fabulous Theology, Part I, Chapter I, published in 1794. He writes:

"As there is now nothing new to be learned from the dead languages, all the useful books being already translated, the languages are become useless, and the time expended in teaching and learning them is wasted. So far as the study of languages may contribute to the progress and communication of knowledge, (for it has nothing to do with the creation of knowledge), it is only in the living languages that new knowledge is to be found; and certain it is that, in general, a youth will learn more of a living language in one year, than of a dead language in seven, and it is but seldom that the teacher knows much of it himself."

"The difficulty of learning the dead languages does not arise from any superior abstruseness in the languages themselves, but in their being dead, and the pronunciation entirely lost. It would be the same thing with any other language when it becomes dead. The best Greek linguist that now exists does not understand Greek so well as a Grecian plowman did, or a Grecian milkmaid; and the same for the Latin, compared with a plowman or milkmaid of the Romans; it would therefore be advantageous to the state of learning to abolish the study of the dead languages, and to make learning consist, as it originally did, in scientific knowledge."

From the mid-nineteenth century onwards, there was greater push to do away with mandatory Greek and Latin. Simon Goldhill, in his essay "Modern Uses of Ancient Greek" in the book The Classical Tradition (edited by Anthony Grafton, Glenn W. Most, and Salvatore Settis, published in 2010 by the Belknap Press of the Harvard University Press) on page 410 quotes the nineteenth-century British government minister Robert Low as having said that it is ". . . more important to know where the liver is situated . . . than to know . . . it is called hēpar in Greek."

There were also nationalist arguments in many "western" countries against the mandatory teaching of Latin and Greek. Some people felt that requiring students to learn dead foreign languages ran contrary to the encouragement of pride in one's own national culture and language. Goldhill notably quotes Kaisar Wilhelm II (ruled 1888 – 1918) as saying "We should raise young Germans, not young Greeks and Romans."

The exact opposite, though, was the case in Greece and Italy. Indeed, the main reason why arguments against the teaching of Greek and Latin have prevailed in the Anglophone world, but not in Greece or Italy, is because, in Greece and Italy, there is an entrenched bulwark of nationalist support for requiring Greek (in Greece) or Latin (in Italy) in schools.

The idea that there is an unbroken line of continuity from the ancient Greeks leading directly to the modern nation-state of Greece is central to modern Greek state ideology, and the same is true for Italy with the ancient Romans. Requiring students to study Classical Greek in Greece and Latin in Italy reinforces the idea of a connection between the ancient past and the present day.

As I understand the situation, the mandatory study of Classical Chinese in the present-day People's Republic of China (PRC) is not totally analogous to the mandatory teaching of Classical Greek in Greece and Latin in Italy, since it is shaped by China's own unique history. In much the same way that knowing Greek and Latin and having mastery of the Greek and Roman classics were considered defining qualities of a gentleman in the west during the Early Modern Period, knowing Classical Chinese and having mastery of the Confucian classics were seen as defining qualities of a gentleman in Ming and Qing Dynasty China.

In China, though, the importance of Classical Chinese and the Confucian classics was, if anything, even more institutionalized than the importance of Latin and Greek was in the west, since, from the Yuan Dynasty onward, the Neo-Confucian canon of Zhū Xī (lived 1130 – 1200 CE) served as the curriculum for the civil service examination system, which a person was required to pass in order to become a government official.

Jing Tsu, a professor of comparative literature at Yale University, argues in her book Kingdom of Characters: The Language Revolution That Made China Modern that, even though the Chinese Communist Party under Mao Zedong strongly opposed traditional Confucianism and the ancient texts associated with it, both the Classical Chinese language and traditional Chinese writing still maintained a very high level of prestige and association with authority, such that even Mao himself practiced traditional calligraphy. As I understand it, in more recent years, there has been a greater burgeoning of national pride in Classical Chinese language and writing in China.