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.
