The Primary president can only make changes to the Primary. The Sunday School president can only make changes to the Sunday School.
But yes, for some of the reasons outlined by eblook66, an auxiliary leader is allowed to make these changes.
Take seriously the training recommendation. A very pertinent ward council training item would be a reminder from the bishopric that changes should not be made at LCR until the sustaining or releasing have actually taken place in sacrament meeting.
In our ward, the clerk enters the calling for a Sunday School or Primary teacher the day they are sustained, and that evening the organization president assigns the member to the class he or she actually will be teaching.