Pesachim 70a - Eating the Korban Pesach While Full

The Mishna 69b explains that a korban chagiga would accompany the korban pesach when there were too many people counted on the the korban pesach. The gemara explains that the korban chagiga was not an obligation and the function of it was only to make sure that the korban pesach was eaten על השובע. Why is it necessary to eat the korban pesach על השובע? Tosafos cites from the Yerushalmi that it is a din D'rabonon because when one is too hungry they would come to break the bones. If people are too hungry, then when they are hastily eating the meat around the bones they will be too rough and break the bones. By implementing a korban chagiga to ensure that everyone would not be so hungry when they came to eat the korban pesach, there would not be a concern that they will accidentally break the bones. Tosafos later on 120a adds an additional peshat - שלא יצא משולחן רבו רעב. It is degrading to walk away from eating a korban and still be hungry. Therefore, we want to make sure that he eats enough before the korban pesach so that he can walk away from the korban pesach satiated. The difference between these two approaches is that according to the first approach we want to make sure that he begins eating the korban pesach על השובע so he won't break the bones, whereas according to the second approach we want him to be full when he finishes the korban pesach. According to the first approach it is difficult to understand why we would only bring a korban chagiga when there are many people counted on the korban pesach, but if there would be only a few, one would not need a korban chagiga. Shouldn't we always encourage one to have many people on the korban pesach so that we can force bringing a korban chagiga to ensure that when they begin eating the korban pesach they won't be hungry? It seems that ideally one should have a large number on the korban pesach so that they are forced to bring a korban chagiga, so that when they begin the korban pesach they are already somewhat satiated and won't come to break the bones.

Rashi seems to have a third approach in understanding על השובע. When one eats something when they are starving, they don't enjoy the taste of it. Only after the severe hunger subsides does one begin to enjoy what he is eating. We insist on a korban chagiga so that one will not be so hungry when they begin eating the korban pesach. By doing this they will enjoy the korban pesach more which is a kavod and chashivus for the korban pesach. According to this approach also it would seem that it is ideal to have more people counted on the korban pesach and force the bringing of a korban chagiga so that when they begin the korban pesach they are already full.