This advice is so spot on. Too often, I try to write a perfect essay from moment one, and inevitably it is all a crazy, incoherent mess. What I have learned, like the author, is to let my brain get it all out in an orderly fashion (linearly, via writing), and that one linear act will start setting up background processes in my brain that are doing processing: filtering, highlighting, pattern matching.
By the time I start a second draft, it is often from scratch, with elements from the first brought in. It is far more coherent and organized, and may move in a radically different direction than the first draft did.
You iterate depending on how important and big this is. For something short, two drafts or so is fine. When I did some legal stuff (not a lawyer, but making some important legal filings) I did many drafts and outlines.
But that act of the first linear writing to get your brain marching is a key first step.