I think there's that component to it which you mentioned with the deep breaths and I also think there's something to say about being in your body while speaking and not in your mind. Deep breathing/cardio/grounding/hard movements really put you into your body which we do not stutter in or very rarely. Even bouncing up and down on feet and grounding yourself, relaxing your muscles, while speaking is very hard to stutter in. Breathwork and bioenegetics is huggeee