The wordlessness of a niggun is the argument, not a shortcut. Hasidic teaching holds that a text fixes what the singer can mean, and that a melody carried on vocables such as bim bam or ay ay ay leaves the intention free to go further than the words would let it. A niggun is therefore not performed and finished. It is repeated, often for a long time, and the repetition is where the work is done, because the melody is meant to move the singer rather than to inform a listener.
Musically the form is modest. Most niggunim are built from two, three or four short sections, each repeated, with the later sections rising in register and intensity before the melody returns to where it began. The materials are ordinary: many of the melodies are recognisably drawn from the surrounding world, from Ukrainian and Polish folk song, from military marches, in one much discussed Chabad case from a Napoleonic march. Hasidic thought does not treat that borrowing as a compromise. The melody is understood to be raised by the use made of it, which is a claim about redemption rather than about musicology.
What distinguishes a niggun from ordinary singing is not its notes but its occasion and its owner. A melody belongs to a court, has a rebbe attached to its origin, and is sung at a particular moment. To sing the wrong court's niggun at a tish would be legible to everyone in the room, in the way that wearing the wrong garment would be, and that legibility is a large part of what the repertory is for.