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.
Other birds, such as peregrine falcons, fly faster while diving in a stoop, but the swift is the fastest accurately recorded flying under its own power. The birds reach top speed during bouts of mating known as 'screaming parties', say scientists.