Chinese, which spread across China 2000 years ago, vary a lot from region to region, even more so than the languages you've mentioned.
After the time of the source, shortly after diverging[1], they would still have been more similar than they are now because they hadn't had much time to differentiate.
[1] Unlike "the time of the source", you can rigorously defend the existence of the period shortly after divergence.
The net difference today is the net difference 2000 years ago plus all the new variations since then.
Why would everyone living in what is now China have spoken the same language 2000 years ago? The people living in what is now the US didn't. Neither did the people living in what's now the EU. Even small territories, such as just the islands of Okinawa had multiple language communities 2000 years ago.
National borders and the language communities within and across them fluctuate wildly over the course of a thousand years.
>" I note Cantonese and Mandarin "dialects" have a common (formal) written form."
They don't even have the same grammar. Cantonese includes characters that don't exist in Mandarin.
You might find this piece on Language Log interesting: http://languagelog.ldc.upenn.edu/nll/?p=548
"Just as Europeans who spoke various languages could communicate in Latin, so until recently all literate Chinese people could communicate in the somewhat artificial written standard. With the shift over the last century to a written language that is closer to the colloquial, the differences among dialects have increased."
That's not really the right question. 2500 years ago, the people in what is currently Spain spoke a language that was distinct from the language then spoken by the people in what is currently Italy. But it's still true that Spanish and Italian sounded exactly the same 2500 years ago.
I don't know spoken mandarin but I can read written Mandarin, I grew up knowing Cantonese. So I do think they have a common formal written form. And I did specify formal, that strips away the slang Cantonese characters.
It's interesting to see the paper compared the written form to Latin.
For nationalistic reasons, China views those as dialects, or did when John DeFrancis was writing about the issue. In the other hand, they share more of a culture (IIUC) and government than Italian and French do.