There are a tremendous amount of laws and customs associated with Pesach. There are eleven different commandments associated with this observance. For example, the Korban Pesach must be roasted over a fire. It can not be cooked in any other fashion.
Roasted meat is a thing of royalty.
It is only fitting that the remembrance we have of how G-d redeemed us from slavery should be eaten in the way that the free and the distinguished eat their meat. Similarly, all of the Korban must be eaten. Leftovers are not permitted. The upper echelons of society have no need to keep leftovers. They know that they will have food tomorrow.