It seems to me that you are offering a much more broad concept of what atemi is in your overall explanation. If so, I understand more of what you are saying. But - I personally would not include "cutting motions" and the like, that simply execute technique, atemi. Many of the body/arm/hand motions that I use to complete the movement of a technique, are sword strike movements. To me that doesn't make them atemi, or that I'm "using" atemi....
If the body and intent behind the movements isn't done as if it was atemi, it simply will not work. All of these movements impact uke's structure, but only if nage performs them as atemi. Most of our transition movements hit to uke's core, even if it's just positioning *because* they are done as if they are strikes.