To throw, we cut. Therefore, our hand, hips and centre and going DOWN. A true throw will only occur if nage is under uke's hips and centre. If this is the case- why do we need to jump over something that lower then we are.
In general, my center had been going up underneath uke for koshinage. The idea is that you throw them up and gravity brings them down. I agree that jumping is a bit off the mark. But I do take control of the movement as the uke. Heck I've throw Gleason sensei who decided to change things up and do a full rotation in the air, land softly on his knees, while throwing me over him. That was just fun - and no I can't describe it much better than that. I have no idea how he did it.