Writing is thinking. The output is the side effect.
People who write well aren't better writers. They've thought more carefully.
Writing is the externalization of thought. A sentence that doesn't make sense on the page is a thought that didn't make sense in the head, made visible.
The act of forcing your idea onto the page is what reveals whether the idea is real or just a feeling. Most ideas die in this process. That's the point.
The writer at the top of their game isn't generating better prose. They're generating better thinking. The prose is downstream.
If you want to think better, write more. Not for the audience. For yourself. The quality of your thinking is downstream of the discipline of articulating it.
Three things I notice when I write regularly:
I catch myself faster. The half-formed thought that would have lived for weeks in my head dies in a paragraph.
I argue with my past self more often. When I revisit a piece I wrote a year ago, the gaps in the reasoning are obvious. That's growth.
I become slower to commit to public positions. Writing teaches you how often the second-best answer is right. The instant-take culture is for people who don't write.
The published essay is a small percentage of the value of having written it.
The bigger value is what writing did to your thinking on the way.