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."