Back to the Queen's Printer, as (once again), nobody at the City of Calgary is inclined to tell us what the Mayor's job is...
General duties of chief elected official
(1) A chief elected official, in addition to performing the duties of a councillor, must
(a) preside when in attendance at a council meeting unless a bylaw provides that another councillor or other person is to preside, and
(b) perform any other duty imposed on a chief elected official by this or any other enactment or bylaw.
(2) The chief elected official is a member of all council committees and all bodies to which council has the right to appoint members under this Act, unless the council provides otherwise.
(3) Despite subsection (2), the chief elected official may be a member of a board, commission, subdivision authority or development authority established under Part 17 only if the chief elected official is appointed in the chief elected official’s personal name.
(Municipal Government Act, Part Five, Division Three)
So, essentially... the mayor is an "Alderman-at-large" required to swing the gavel at meetings.
Again, as with Aldermen, the Mayor can do much more if he (or she) wants to get re-elected. Ralph Klein, when he was Mayor of Calgary, focused on the task of communicating with the people that the council served. Bronco seems to view the mayor's role as more of a leader, helping to set the priorities of council as he believes the people of Calgary wish them to be. Each mayor, and each candidate for mayor, would bring their own interpretation of what the role actually entails.
As long as they've got good gavel technique, none of them is wrong... but some are more right, in the eyes of the voters, than others. Therein lies the difference between "Mayor Such-and-such" and "defeated mayoral candidate Such-and-such".