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.
TRACK, BOWLING, ROAD, and POOL all fit the category "Things with lanes". Movement is organized into lanes. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.
Category
Things with shells
EGG / TURTLE / ACORN / CLAM
EGG, TURTLE, ACORN, and CLAM all fit the category "Things with shells". Each can have a protective outer layer. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.
Category
Board game pieces
DIE / PAWN / TILE / TOKEN
DIE, PAWN, TILE, and TOKEN all fit the category "Board game pieces". You might move or roll these on a tabletop. 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
MAT / KETTLEBELL / BENCH / ROPE
MAT, KETTLEBELL, BENCH, 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.