Here is the context of St. Augustine's remarks: "You ought to say plainly that you do not believe the gospel of Christ. For to believe what you please, and not to believe what you please, is to believe yourselves, and not the gospel."
No, this is the context of the statement: "But if believers can now say what is true of Christ because the truth has been handed down in word or writing by those who saw and heard, why might not Matthew have heard the truth from his fellow-disciple John, if John was present and he himself was not, as from the writings of John both we who are born so long after and those who shall be born after us can learn the truth about Christ? In this way, the Gospels of Luke and Mark, who were companions of the disciples, as well as the Gospel of Matthew, have the same authority as that of John. Besides, the Lord Himself might have told Matthew what those called before him had already been witnesses of."
It is the scriptures Augustine is pointing not the unwritten tradition that you are alluding to.