I think you are correct in your assumption that stuttering has a cognitive effect, however, I don't think it is due to faulty breathing as you stated. During a stressful situation, the brain shuts down unnecessary functions, including cognitive reasoning and critical thinking. The brain will literally switch from being intelligent, to focus only on survival. If you observe your thoughts patterns during a stutter, you will realize that you are thinking of an "escape" from this speaking situation, e.g. anticipating blocks, replacing words, over formulating your sentences, .... etc. If you are doing this while expressing an idea, your brain while get stuck in this thought loop, and it would be hard to escape. The brain cannot multitask in thoughts, you can't think of a complex idea while simultaneously think of the shortest way out of a conversation. It is physically impossible.