unified writing system apparently contributes greatly to the unification and cohesion of nations and peoples

Does it? Genuinely curious. I know this theory of Chinese unification's around for like ever, but Ancient Greek, Latin, pre-modern German, and pre-modern Italian did not seem to have much affects (though not my area of specialization of course so I could be wrong). Modern Spanish also don't seem to be doing anything to unify Latin America. And a unified written language did not prevent China from splitting apart time and again. And of course there's the fact that in the pre-modern the vast majority of the population can't read and write. Perhaps a unified written language is not a cause but a product of unification that has to be constantly maintained by a political power. Do we have any Latin/Greek experts who can answer that?

Since a unified writing system apparently contributes greatly to the unification and cohesion of nations and peoples,

What's your basis for this claim? You'd first need to show that the premise is true. Are nations made more harmonious with a shared script? There are plenty of counterexamples.

You also need to ask if Japan was not this way prior to the introduction of a written language, but since we lack records on that state, I'm not sure you'd be able to. The introduction of Chinese writing was the introduction of writing period, and there were plenty of major periods of disunity after that point.

What's your basis for believing the premise to be true?

I've never heard about chinese script having a role in unifying Japanese society soon after being introduced, but language and state is actually a common topic when it comes to the creation of the modern Japanese state in the Meiji period. But still, it was more about the creation of a unified spoken language which did not exist at the the time, and the unification of spoken and written language (which was very different from modern spoken language). Japan had (and still has) a strong dialect divide, until the early Meiji period the role of a common language for communication was covered by the Kyoto variety, until the center of power definitely shifted to Tokyo. Only then the Tokyo variety of japanese starts to be considered the common language for communication. The idea of a unified language for a unified state in Japan was mainly introduced by the linguist Ueda Kazutoshi (or Mannen), who studied modern linguistics in Germany and France and was influenced by european linguists. The Japanese government eventually formed the Language investigation committee（kokugo chousa iinkai国語調査委員会）, which amongst others had to investigate japanese dialects and the possibility of a common unified japanese language, the unification of script and spoken language (genbun itchi 言文一致) and phonetic scripts. As a result of their investigations the first standard spoken language textbook was printed, "Rules of spoken language" (Kougohou 口語法), based on the language spoken by the educated elite of the Yamanote area of Tokyo. Language education in schools, together with media diffusion and other social changes in the later years is credited for the diffusion of the standard japanese language (hyoujungo 標準語). We can't really say if language policies like these unified Japanese society, probabily they did contribute, but what is true is that when the modern state was born in the Meiji period the need was there and language policies shows for this.

Sources

Sanada Shinji, 2001, Hyoujungo no seiritsu jijou, PHP bunkou

Yasuda Toshiaki, 2006, Kokugo no kindaishi - teikoku nihon to kokugogakushatachi, Chuukou shinsho