Whether they have any extant function or not, they have lost their former function and in that sense they do fit the definition of vestigiality. Vestigial structures are often called vestigial organs, although many of them are not actually organs.
There are quite a few birds famous for doing ritualized mating dances, so I would only list 5: The Red Capped Manakin, The Black-footed albatross, The Western and Clark's grebe, The Superb bird of Paradise, & The Victoria's riflebird.
The first part of this paper will establish the near universality of a symbolic connection between birds and death. The following sections will provide sev- eral examples of each type of association, drawing on the folklore, supersti- tion, and tradition of cultures spanning the world throughout history.