What does it mean to write? How does it help? Let me list the three things that come to my mind most easily.
Writing frees up my memory. It keeps me from constantly juggling with information in my head. This applies to shopping lists, to-do lists, as well as to problems and concerns that plague my head. Writing helps imprison them on paper, or in more recent times, behind a screen. It helps me look at these items and prioritize them. Whatever festers as a demon in my head eventually turns into a mere trifle after I write it down.
Secondly, it helps me step above information and observe the patterns that emerge. Journals, blogs and articles help their authors ideate by doing so. Along with this clarity comes accountability. I am forced to think and express my ideas coherently, since it is impossible to take back what I have written.
Most importantly, writing gives us feedback on what our thoughts look like. Unlike spoken words, which fade away, written words stare back like reflections from a mirror. Thereafter, writing enables us to go back and revise what we express, closing the loop. By editing our sentences, we are also rewiring our brain.
Writing is one of our oldest triumphs as a species. At the same time, it is sorely underrated. Only a handful of people utilize it. There is scope yet for a writing revolution – where each person writes everyday just as surely as they bathe or brush their teeth.
Observations 
I think paragraph 4 is not very accurate.
Most people do harness the power of writing. For instance as you said in paragraph 1, a lot of other people also make shopping lists. Writing is used to capture meeting minutes, it is the preferred form of correspondence in the professional world (even more so than talking) or even to do finite element analysis.
All of the above is done only by writing
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Arjun, thanks for your comment, and you make a valid point. Most of us use writing, but only a handful utilize it. To utilize is to make practical and effective use of something – using it to its real potential.
I think writing is a skill that ought to be a part of our basic education, with its effective use ingrained as a habit.
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