At least according to Koichi Tohei Sensei, use of "kokyunage" is meant more to rhythm or timing.
breathing is rhythmic and that ("rhythm") is the sense you should be interpreting the name.
kokyudosa is another case where I think people are being misled by literally interpreting the name.
I very much agree with you in regards to rhythm. However I don't think people are misled by the name. Like pretty much everything in aikido there are multiple issues at play here. I am merely talking about one aspect.
To employ the principles laid out by Tohei in an example, it's like I'm talking about relaxing. If you counter with, but Tohei says we should do the technique with weight underside, you're right but you haven't invalidated my point.
Natural rhythm is also an important and integral part of a kokuynage. The challenge then, as I see it, is to sync the breath of the motion to your own natural movement, with the motion of your partner. This is a heck of a lot to deal with. This is another reason why I view this idea as an advanced level concept.
It's important that we understand both of these components and not get too wrapped up in one or the other. If I argue breath to the exclusion of rhythm it's like arguing an emphasis on ki to the exclusion of posture. Look at Tohei. He's spent his life spreading an understanding of ki. When I think of Tohei, I think of ki principles. Posture is not one of the first tenets that pops into mind. However, if you've ever seen him, his posture and precision is incredible. He obviously understands that each is a component (hence the principles).