This is funny, perhaps like others here, I find I usually have to reduce the complexity of my speech compared to my internal dialogue: particularly in using more commonplace vocabulary. Sadly barely anyone I know (including my wife) understands half the words I use. Indeed sometimes I don't really grok them myself; my brain just finds them most apposite.

So should I write as I [find I need to] speak or should I write with my natural lexicon?

Depends. The main point the author in the OP doesn't seem to understand:

Know your audience.

"Dumb down" your writing, but only if it will help you communicate with the person (or people) you're targeting. Else there's no benefit to it.

I think so, but I was more thinking about what people actually do when you tell them "write like you speak." Most people I've seen then proceed to dump stuff onto the page and it's even worse.

The advantage of reading it out loud is (I'm guessing) that you process it with a different part of your brain. When you just read it, the words are your own and you don't process it the same way, whereas when you hear it the words in a sense sound external and it is easier to hear where it sounds awkward.