Systema -like many other arts- doesn't say or tell everything. If you don't know an advanced teacher, I wouldn't assume their breath work is all covered.
Systema breath work runs deep- far past relaxation.
As for Kokyu and falling you can also do Systema's version of breath work to gain heavy hands in their pushup as well. But I wouldn't say you need to "fall" on your hands to learn that skill. There are other ways to get the feet-into-the-hands. I could also argue for training ground-work and how to use the body and here breath work can be an even greater advantage. But it still is not ukemi is it? The best way to train Kokyu skills is standing. I will argue-nicely of course-that the best way to train standing is to train? To remain standing.