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.
SPRINTER, COMET, ARROW, and FALCON 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
Card suits and kin
HEART / SPADE / CLUB / DIAMOND
HEART, SPADE, CLUB, and DIAMOND all fit the category "Card suits and kin". A deck has all four. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.
Category
Things that fly
EAGLE / JET / DRONE / KITE
EAGLE, JET, DRONE, and KITE all fit the category "Things that fly". They spend time in the air. 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 / SCENE / ENCORE / PROP
CURTAIN, SCENE, ENCORE, and PROP 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.