I don't think a bifurcation of cognition is necessary to explain this. You get both people who claim to always think with words (to the point where they don't understand how it's possible to think without one), and people who claim never to outside of linguistic tasks like writing (to the point where they need "internal monologue" explained to them). In my experience, if you get down to it -- most people will have experienced cases of the thought preceding the word (you can skip the word, or not, but either way it demonstrates a thought without a word) and of the word preceding the thought (people who claim not to have an internal monologue will usually still do this sometimes when writing).

It's just that subvocalization is one powerful technique for organizing thoughts, but not the only one. If you had a swimmer who only ever learned the crawl stroke (freestyle), they might wonder how someone can swim without it. The difference, of course, is that the breaststroke is a demonstrable physical act whereas cognitive techniques are not. The monologuers are just more practiced at subvocalization-based reasoning, and the non-monologuers are more practiced at other types.

It's not that hard to exert limited control over which technique you're using -- and I think there are multiple nonverbal techniques. I'd encourage people to explore the space a bit. Subvocalization-first approaches are very good for linear progressions, whereas subvocalization-last (or never) approaches are usually better for big-picture reasoning.