Not sure whether I get you right.
But I think you describe what we teach as the most basic form of ude kime nage:
atemi to throat or face with the arm which delivers the kime to ukes arm. Stretching uke. Then atemi to the genitalia with the same hand. Dropping uke.
Yes, that is what I was referring to. I think you gave a more articulate and concise description than I did.
Whether you are going for the harder 'atemi version' or the softer 'kokyu version', I think the trick is not to worry about the elbow so much. If you move correctly, the elbow will take care of itself.