Unless authorised to do so by the council, a councillor has no authority to speak on behalf of the council which is a corporate body. Absolutely nothing to prevent anyone speaking as an individual on behalf of residents if they so wish but they can't say they're speaking on behalf of the council. Policies "which aim to prevent Councillors doing this" should clearly indicate this.
One word of caution however is that speaking out as an individual on a subject that the council will consider (I'm thinking planning but there may be other matters) may lead to a challenge on the grounds of pre-determination when the matter comes to council for a discussion/decision.