> Mandarin is not written Chinese to my knowledge. […] Source: I’m native Chinese
You are confusing written language with writing system, probably due to domestic and global propaganda efforts that have been going on for 70 years. This really undermines the scientific effort of linguists and just sows confusion, as evidenced by your mislabelling languages as dialects, and forcing the rest of the world to invent new words like "topolect" for existing concepts that the communist party does not like to be true.
My writing system is the Latin alphabet. I can write the languages Danish, English, French, German, Hungarian, Italian etc. etc. with it.
Another writing system is the Cyrillic alphabet. I can write Bulgarian, Russian, Serbian etc. with it.
So the writing system is an orthogonal concept to the written language.
As an analogue, the writing system we are discussing is the Chinese characters. One can write with it for example Cantonese, classical Chinese, Hokkien, Japanese, Mandarin, Zhuang, among others.
The distinction between traditional and simplified you mentioned is minor and not a fundamental property of the language written with it. Example: Cantonese can be and is written in both simplified and traditional.
> the written form is universal
That's not true. The languages Cantonese, Hokkien, Mandarin belong to the same family (Sinitic) but have different grammar, lexicon and word order; these differences are also reflected in writing.