Hi NH. It's OK that you posted here. Folks will still see it and share whatever they can.
What, if anything, do your bylaws say about chairmen, how they're appointed and what role they play in the PTO? For example, I worked on a team who's bylaws said:
The President shall appoint all committees and shall designate a chairman of such committees subject to Board approval.
This basically says the president can pick whomever they choose but then the board has to approve the appointment. If the board says no, the president offers up another candidate and continues to do so until the board is in agreement with the appointment.
This particular group had an unwritten policy of offering next year's position to the current chairman. It wasn't, however, mandatory. A person could also seek a chairman position by asking the president for it. However, no person,
ever, could demand the chairman role.
As for how "Mary" should be approached, I agree that it should start with you meeting with her. A non confrontational meeting to discuss the chairman position is a good first step. Simply say, "Jane ran the committee last year and did a wonderful job. She would like to run it again this year and the board has agreed to keep her in that position." If she continues to demand the position, then set up a meeting with her and all of your board members. Let her know that you recognize Jane as the committee chair. Then, perhaps, in the spirit of teamwork and inclusiveness, offer something else to her.
Finally, regarding the grievance she has with the teacher, don't let it get discussed. The only place she should even be able to present the topic is under New Business. As soon as she does, say that it is not a PTO issue. Then move on to the next New Business or agenda item. It really is as simple as that.