I know these are usually safe ways of practicing, because throwing someone on their face is hard to train safely. I'm wondering why it was so important for knights, legionaries, samurai, hoplites etc. to get their opponent on their backs, since they've all trained grappling in similar ways.
The answer to this is rooted in physics more than history. When an opponent is on his back it is much harder to gain an advantageous position. As the person on the ground you have to continue to fight aggressively while your opponent only has to deflect your blows and put his weight on you, allowing gravity to aid him in keeping you down. As you continue to struggle to get out or fight him off of you there is a lot of energy lost and soon muscle fatigue will set in. Meanwhile he retains his strength, having to use only a fraction of the energy to maintain his position as you are having to expend defending yourself and trying to get away. Once you are exhausted it is a simple matter to dispatch you.