The license trustee is responsible for the proper operation of the repeater.
As others have said, "it's his license on the line".
Repeater control codes (at least sufficient to turn off the repeater transmitter in the event of a problem) should be in the possession of the license trustee, and any designated control operators.
Now, if the repeater is owned by a club, the license trustee, and any designated control operators take their marching orders, within the legal confines of 47CFR97, from the elected club officials. But they still need to be able to control the repeater, which means they still need to have the repeater control codes.
73 de N8AUC
Eric
PS - Yes I am the license trustee for our club's repeaters.