I do agree that writing is not _necessary_ to improve your thinking. However, I think that writing about a topic will often if not always improve your understanding of that topic as well as your ability to communicate it.
I think that's because when writing about something you have to be slower, more deliberate and more structured than thinking about something in your head which helps you identify gaps in your understanding which you can easily miss when the idea is in your head.
I think the better way overall though is just to try different methods of thinking and do what works for you. I also think more than one method is almost always better than just using one.
I find that writing and mathematics are just a poor way of trying to structure a context free domain specific language for the problem.
Writing is bad because it needs to fit in a human head and ultimately needs to be spoken by a human mouth. Maths is bad because it is still stuck as being written on blackboards or pen and paper if you're unlucky.
The few times I have build up these dsls formally have let me understand a topic both more deeply and remember it more easily. The downside is that time invested in inventing it in the first place. E.g.
(for-all x (implies
(member x integers)
(exist-unique y
(and (member y integers)
(equal? 0 (+ x y))))))
Is the way to say that all integers have additive inverses. It's ridiculously long but logically sound and quite easy to prove mechanically given that every operator in that sentence also have definitions in that vain.
I think the better way overall though is just to try different methods of thinking and do what works for you. I also think more than one method is almost always better than just using one.