This difficulty may be resolved by saying that there they are dealing with a public fast-day, as we learn in the Mishnah in Ta'anit (16a) that we appoint only an elder who is familiar with the prayers, and one with much responsibility.
And as to what it teaches there with respect to one who has a full beard being worthy of being appointed the representative of the congregation --that is referring to a permanent a permanent prayer leader. However for a one-time occasion it is sufficient that he reach puberty.
This explanation disagrees with that of Rashi's commentary there, since he explained that they were speaking about the sounding of the shofar, or of an appointment to a position of community leadership, with authority to order lashes or excommunication.