They are elected by the Board of Directors.

That is to say, by Mrs.Eddy.
Their terms of office expire on the first Tuesday in June of each year, "or upon the election of their successors." They must be watchfully obedient and satisfactory to her, or she will elect and install their successors with a suddenness that can be unpleasant to them.

It goes without saying that the Treasurer manages the Treasury to suit Mrs.
Eddy, and is in fact merely Temporary Deputy Treasurer.
Apparently the Clerk has but two duties to perform: to read messages from Mrs.Eddy to First Members assembled in solemn Council, and provide lists of candidates for Church membership.

The select body entitled First Members are the aristocracy of the Mother-Church, the Charter Members, the Aborigines, a sort of stylish but unsalaried little College of Cardinals, good for show, but not indispensable.

Nobody is indispensable in Mrs.Eddy's empire; she sees to that.
When the Pastor Emeritus sends a letter or message to that little Sanhedrin, it is the Clerk's "imperative duty" to read it "at the place and time specified." Otherwise, the world might come to an end.