Whether the Didache and the writings of the ancient fathers are correct or not would depend on when they were written and by whom they were written. Anyone can write about what they believe to be true. The Pharisees wrote many laws they deemed to be from God.
The Didache was written in the mid-to-late 1st century. The word Didache means "Teaching", and is short for "Teaching of the Lord through the Twelve Apostles to the Nations". The Didache provides for us a kind of manual for instruction and teaching which would have been used for catechetical purposes in the nascent church. That is, it is the earliest catechetical tool we have, providing for ethical instruction, on how to conduct the sacraments, church discipline concerning traveling preachers, and a reminder about the end of the age.
The writings of the early fathers are well attested. We have the Epistle of Clement written by St. Clement of Rome, St. Peter's successor as bishop of Rome, writing to the church in Corinth around the year 90 AD; and possibly the same Clement who was one of St. Paul's missionary companions. We have the Epistles of St. Ignatius of Antioch, these being addressed to the churches in Ephesus, Magnesia, Tralles, Philadelphia, Smyrna, and Rome, as well as a letter to Polycarp the bishop of Smyrna. Ignatius, the bishop of Antioch, who was an old man at the time he wrote in 107 AD as he was being led in chains by Roman soldiers from Antioch to Rome in order to face his inevitable martyrdom. Ignatius had personally studied under St. John the Apostle, and succeeded Evodius as bishop of Antioch after Evoidus' death in 66 AD. Polycarp, who was close with Ignatius, was bishop of Smyrna and was one of the original recipients of the Apocalypse (written to the seven churches, one of which being Smyrna). He likewise wrote a letter to the Philippians, sometime between 110 and 140 AD, and around 150 AD suffered martyrdom by being burned alive. The record of Polycarp's mockery of a trial has him declaring boldly even in his advanced age at the time, "eighty and six years I have served Him, and He never did me any injury, how then could I blaspheme my King and Savior?" (note, here, that Polycarp who was 86 years old at the time of his death, say he has been a Christian for 86 years).
Likewise we have the writings of St. Justin, including his defenses of the Christian faith against the false claims of the Pagan accusers, directed to "the Emperor Titus Aelius Adrianus Antonius Pius Augustus Caesar, ... and to the sacred Senate, with the whole People of the Romans," Justin is named Justin Martyr because he, like so many others, faced the end of their days in martyrdom.
Further, we have St. Irenaeus, who became bishop of Lugdunnum (modern Lyons) in southern Gaul, had studied under Polycarp, and was acquainted with Polycarp and Ignatius' familiarity with the Apostles. Irenaeus wrote a massive tome against the Gnostics, a five volume work known as
Against the Gnosis Falsely So-Called, aka
Against Heresies. In this massive work he defends the apostolic teachings as they had been received from the beginning, having been safeguarded by men chosen by the apostles themselves, and publicly proclaimed and confessed in all the churches. He, likewise, attacks the false doctrines of the various Gnostic heretics for their obvious charlatanism and grift.
Along with Irenaeus we have St. Hippolytus of Rome, to whom the Apostolic Traditions are attributed, who along with Irenaeus had also been acquainted with Polycarp. Hippolytus, in addition to the Apostolic Traditions attributed to him, wrote a refutation against heresies, a treatise against the heretic Noetus (one of the advocates of the Sabellian heresy), as well as exegetical works which are now only known in fragmentary form.
Additionally in this period we have the works of Melitio of Sardis, Athenagorus of Athens, Theophilus of Antioch. We also have Tertullian of Carthage, but whose works are marred by his involvement with the Montanist heresy. Among many others. And this is only covering the period between the mid-to-end of the 1st century to the end of the 2nd century. Not yet even discussing the major figures and important work of those of the 3rd century and later.
-CryptoLutheran