I think “the writing is the thinking” is just combination of selection effects and typical mind fallacy. Of course people who contributed a lot of writing to the Internet are going to be the sort of people who put large weight on writing in their cognitive process.
the act of forcing yourself to write that down into sentences and paragraphs to structure the argument will show you where the holes are in the argument
You should look for both conditions of presence and absence of phenomena. Yes, writing forces me to think, but I spend a lot of my waking time thinking about abstract problems and staring into the wall pondering my thoughts works just as fine. If anything, to write I should stop thinking, because it’s never the case that I finished my understanding and in some moment I should write whatever I have on my mind to have something written.
This is not to undermine the rest of your thesis (LLMs are bad at writing and they are unfaithful in conveying complex thoughts in my experience), but “the writing is the thinking” always sounds to me like an assertion of typical mind fallacy.
How is the progress six months later?