Brinker is under the impression that Gene knows more than he is letting on about Finny's fall from the tree. The reason Brinker calls the meeting is to bring the facts to light.
Brinker and "three cohorts" barge into Gene and Finny's room and force them to proceed to the Assembly Room in the First Building.
His suspicions about Gene do not extend to any kind of certainty that Gene caused Finny's fall. In fact, it is the lack of clarity that propels Brinker to call the meeting/inquiry. His character is defined by a need to be proactive and to eliminate doubt. He cannot abide any doubts to remain intact regarding Gene's role in Finny's fall.
If Brinker knew that Gene was involved somehow in Finny's fall, with certainty, he would not call the meeting and would have no need to call witnesses (Leper).