This is the key bit. It's not so much "write like you talk" and more "write like you could present it."
So much writing isn't accessible because the author misses this cardinal rule. Even the most dense, formal scientific writing can be made clearer if the author or editor reads it aloud and fixes the awkward parts.
I helped content editing for an academic journal back in the day. A big part of our jobs was helping the authors bring their voice out into the papers. A lot of writers, novice and expert alike, have an idea of what an "expert" in their field should sound like. The result is stiff and manufactured prose that too often ends up being inaccessible for anyone else.
But, as soon as you clue an author in on how their personal voice can come through an article, you see a shift in the quality and clarity of the work.
The articles, text books and other writing I learn the best from, and come back to, ultimately sound in my head as if the author and I are speaking aloud. It's the real differentiator in sounding like an amateur or an expert.