For the council to function there must be a proper officer, whether paid or unpaid. If it is a councillor (or an ex-councillor within twelve months) they cannot be paid. There must be a proper officer for each major function, whether that is a single person or more than one person.
Nobody can be a proper officer unless appointed by the council in a properly constituted meeting. Provided there is a proper officer for the main functions, the council can continue indefinitely.