I think the only thing you could do as an FC leader now who wants to leave chest access mostly open is to have a "contract" in place. For example have all new people come in at a rank with no access what so ever and have them in a tell agree that they are not allowed to clear out the chest and leave. If they do so and you keep a screenshot of the tell you can report them for breach of agreement.
SE already enforces these agreements for gil loans of if you ask someone to make an item for you and you give them the mats. I don't see why this would be any different.