Some of the songs at the end of the seder have an obvious connection to what has come before. Adir Hu is about praising God, which we've been doing all night; Eliyahu HaNavi makes sense because Eliyahu is part of the seder; Bashana Haba'ah fits with our desire to be in Eretz Yisrael next year. And then there are the two progressive songs, one about a goat and one that counts up various important things, none of them about Pesach in particular. Why were these songs chosen for the seder?
I am aware of this similar question about Echad Mi Yodea. The comments and answer there are basically "to keep busy", but that doesn't answer the question of why these songs in particular.