Written Chinese could arguably be considered its own language.

Sure, by someone other than people who scientifically study human language, for the reasons outlined above. The study of orthography is its own separate (though closely related) field for good reason, though it’s nowhere as big as linguistics, since it’s not as scientifically interesting.

There are several spoken languages in China which are unintelligible to each other, but that look the same when written down since the written language doesn’t codify phonemes or even spoken words, but concepts.

You do understand why this supports my argument, right? Writing is just a largely arbitrary system of (imperfectly) encoding/representing human language, which must be learned, and is not acquired the way human language is. For this reason, it makes perfect sense that what is effectively a “code” for language could be used to represent multiple languages. You could just as easily do the same with written English. Heck, formal logic is specifically designed to do this for all human languages, but that doesn’t make it a language itself.

Here’s a pop article talking about the distinction, reflecting the discussion above (spoilers for the movie Arrival, which I highly recommend if you haven’t seen it). I can’t point you to any peer-reviewed articles on the subject, of course, because this has been decided science since the publication of Ferdinand de Saussure’s 1913 Course in General Linguistics.

I hate referring to Wikipedia (again however, there are no articles on this because it’s settled science), but note that the article for Writing system correctly identifies writing as representing human language and not actually consisting of it.