When is it better to think without words?

Problem solving is a well-explored field in experimental psychology. TFA is a bit unfocused, making both some generally supported speculations and some traditional ideas that haven't been supported. A very good survey is the edited volume, The Psychology of Problem Solving (Davidson 2003).

Although TFA doesn't refer to it by name, "insight" problem solving is when you are stuck on something and then suddenly realize the solution. The common explanation for being stuck is "fixation" on the wrong things. In agreement with TFA, there is indication that verbalization supports fixation more than visualization.

It's quite frustrating when writers like the author - who seem quite thoughtful and potentially useful to read - appear to pontificate on a field without seemingly being aware of it; using a great mathematician's thoughts on it as a springboard doesn't justify it as its not his field of expertise either.

The essay might be more useful grounded with references to the sort of thing you link to.