Unfortunately, your impression of the relationship between spoken and written varieties of Chinese is not entirely correct. This brief answer I did a while back, with links to answers by /u/keyilan, goes into this in more depth.

/u/EnclavedMicrostate already hit the main point with the premise but I wanted to expand a bit.

So I was thinking of the Chinese situation where you have a number of highly divergent dialects … using the same writing system.

So, we can say that yes they do use the same writing system, in the sense that French and Vietnamese use the same writing system, i.e. a script derived from the Latin alphabet. That says nothing of the linguistic similarity, only that the basic set of components is the same.

However, just in case what you had in mind is that they are all written the same, and not just about the use of characters, then of course that's not the case, and that's what EM's answer gets at. Written Mandarin is not the same as written Hakka which is also not the same as written Cantonese, even though yeah there are some similarities and some of the words are the same. But not all, and not all the words are anyway, and anyway the grammars also differ. It just so happens that in many of the places where non-Mandarin languages are spoken, there's a situation of diglossia — i.e. a functional distribution of two or more language varieties in a single community — where Mandarin simply is the written language regardless of what's spoken. In that regard it's quite like Latin (or Greek earlier on) in the Roman Empire, where you speak one language and write a completely different one.

I'll let one of the Rome experts get more into the details of the situation with Latin.

… [dialects] or languages I have no dog in that fight …

Indeed there is no fight. In general the only people wasting any time arguing about this non-issue are non-linguists. Among people who actually engage in the scientific study of linguistic varieties, this literally never comes up.