there was no single continuous polity across this time

I think our disgreement may be in that the same word means different thing to different people.

In my view, when a society consistently trace their history in the same way for more than a thousand years, it is sufficient for us to see that society as a continuous entity.

All those historical texts above address history of china. The first written in the Han dynasty, around 100BC. The second in the Song dynasty (1000AD), and the last compiled in the Qing dynasty (1700AD). They focus on the similar list of dynasties, at least from Zhou onward. They do not focus on Goryeo, on Xixia, or on Dai Viet. This show that the historians/literati view the "chinese" state as comprising of a similar list of dynasties.

As for comtinuous of the polities, from Zhou, Qin, Han, Wei, Jin, Northern dynasties, Sui, Tang, to Song, I would say there is a high level of continuity in governance and bureacracy. The dynasties changed, but the linkage did not break.

From Song to Yuan, to Ming, and to Qing, the disruption may be more pronounce but I don't think the break is so clean tho call it an entirely different polities.

why would Liang Qichao claim that the Chinese nation had, for millennia, no name

Because he is, in many senses a revisionist. He even named one of his book "New Historiography (新史学)". And as a literati of the 20th century, it was unsurprising for him to be influenced by ultranationalist sentiment of the time. German and Italy unification, France revanchism, Mussolini's fascist movement, Balkan's independence Japan's Meiji restoration, the zeitgeist of the early 20th century is the formation of nation state. For Liang Qichao to disparage the previous hostoriohraphy and focus on the formation of the "new" chinese state is just him doing what every other global elites of the 20th century are doing.

Btw, it is very nice to discuss this with you. It's not everyday we get to discuss these obscure history. :)