All of those things point to rhythm in my opinion, which is also what I would define Aiki to be all about.
It's often said when something is "knacky" that it takes a certain rhythm. Breathing is super rhythmic, time and tone are at the core of rhythm. I don't know about secret.... Maybe added by someone who doesn't understand rhythm (Ha).
In an interview by Meik Skoss of Sawada Hanae, in Koryu Bujutsu, Classical Warrior Traditions of Japan. He asked her about aikido technique, she said: "When you speak of matching, if your kokyu, or breathing, does not match then you do not match. You are thinking merely of form, aren't you? But in order to do aiki, both your spirit and that of your partner must enter into play and then come together. When you study aiki, this is what you are studying."
Kokyu seems to be an integral part of Aiki(do) from this.