The full solution is below, with a short explanation for each group. To avoid spoilers, play the puzzle or open the hints page before reading the answer panels.
KITE, PUPPET, HOODIE, and VIOLIN all fit the category "Things with strings". A string is part of the object or how it works. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.
Category
Stage terms
CURTAIN / PROP / ENCORE / SCENE
CURTAIN, PROP, ENCORE, and SCENE all fit the category "Stage terms". A theater audience might notice them. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.
Category
Fast things
FALCON / ARROW / SPRINTER / COMET
FALCON, ARROW, SPRINTER, and COMET all fit the category "Fast things". Speed is the common thread. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.
Category
Film crew roles
GAFFER / DIRECTOR / EDITOR / PRODUCER
GAFFER, DIRECTOR, EDITOR, and PRODUCER all fit the category "Film crew roles". They appear in movie credits. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.