The "China is 5,000 years of unbroken civilization" trope is also a bit of historical myth-making; aside from the obvious fact that the modern Chinese state is very young, "China" has, much like Europe, been a churning amalgamation of numerous different civilizational centers and cultures, with going periodic "foreign" influxes, back to the earliest periods for which we have evidence. The idea of a unitary, persistent, continuous, and homogeneous "Chinese civilization" is a relatively modern, ideological position, that tells you something about the people who bear it (in the same way that someone conflating modern 'Western civilization' and ancient Greece might be signalling something about their ideology).
There's much more continuity of the idea of China than you're suggesting. The existence of a relatively stable writing system, such that people today can still read ancient texts with some practice, of unifying ideas like the Mandate of Heaven, of Legalist governmental practices and Confucian values have created a much greater level of contiguity than in Europe. Even the foreign rulers were Sinicized to a great extent (similarly to how Rome, conquering Greece, was conquered culturally by the latter).
In all seriousness, it seems that people still do value the ability to write characters beautifully - much more than people value handwriting skills in the United States, at least.
