Within Seidokan my teachers taught me that "breath" wasn't quite correct as a translation of kokyu. They used "timing" vastly more. Moving when it is "right". So breath wasn't emphasized (by my specific teachers, that is, or maybe better yet that wasn't what I took from it). Instead, it was about timing the throw/movement/whatever so it happened "at the right time". One day someone commented that you it's like timing a wave. At certain points in the wave you can go up, others you can go down, others are neutral. All are possible but each at the right time. I remember another person commenting that it's also "like" breathing (as in analogous, not due to). You exhale when the time is right. And that coordinating your breathing with your movement is important to moving well. But I didn't take from that anything about using the breath per se. Just that things are supposed to be "in sync".
And keep in mind that all this is what I took from it, not necessarily what they meant.
I can be a bit dense sometimes...