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.
PROP, ENCORE, CURTAIN, 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
Kitchen verbs
WHISK / DICE / FOLD / SEAR
WHISK, DICE, FOLD, and SEAR all fit the category "Kitchen verbs". These are actions from a recipe. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.
Category
Things in a gym
KETTLEBELL / BENCH / MAT / ROPE
KETTLEBELL, BENCH, MAT, and ROPE all fit the category "Things in a gym". Training rooms are their habitat. 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
KITE / EAGLE / JET / DRONE
KITE, EAGLE, JET, and DRONE 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.