Yes, I believe that CALC is wrong, a view supported by the NALC, whose guidance states:-
"The membership of a committee is determined by the council, and the chair of a committee may be elected by the committee from among its members or is elected directly by the council when the committee and its members are first appointed. Some councils may want the chair or the vice-chair of the council to be members of a committee by virtue of their office (i.e. ex-officio) rather than because of their nomination and/or selection to a committee, as would be the case with other members of a committee. A council’s standing orders may require ex-officio members to be full voting members of every committee appointed. Alternatively, as suggested in paragraph 9.54 on page 223 of the book ‘Knowles on Local Authority Meetings – A Manual of Law and Practice, 8th edition’, a council’s standing orders may permit ex-officio members to be members of one or more committees but with restricted rights in respect of such committees. Such standing orders could restrict the ex-officio members’ rights to mere attendance of committee meetings. The wording of standing orders must confirm the rights of ex-officio members of a committee if these are different to other members of the committee."
The final sentence suggests that if your standing orders do not restrict the rights of ex-officio members, they have the same rights as everyone else. This is certainly the case for charity trustees and local drainage boards. It is disappointing that the model standing orders do not mention ex-officio appointments, as this would help to clarify the position.