Actually, the biggest knock against the Didache is the section on dealing with itinerant prophets – which was very time and place specific, dealing with an issue which was not universal.
Apostolic authorship is a good “rule of thumb” but is not, by itself, a defining factor. For example, the author of Hebrews is unknown, yet it is included, while Clement is excluded, though he was a disciple of both Peter and Paul, and is mentioned by name in one of Paul’s letters.
Coming from the apostles or one of their students has nothing to do with it. Justin Martyr was a student of the apostles but he isn’t in the Bible. For all we know the Apostles may not have even written the gospels, yet they are there. Also, the Didache’s style is supposedly similar to Matthew, leading some to believe it has the same author.
Mark Shea argues pretty persuasively otherwise in his book
By What Authority? His argument, which I think is accurate, is that one of the hurdles which had to be overcome was Apostolic origin – not that it had to be *written *by an Apostle personally, but had to have been derived from an Apostle. So Mark’s Gospel is acceptable because he was Peter’s disciple, Hebrews was written either by Paul or one of his disciples (Clement, Barnabas, etc.), Luke’s Gospel is acceptable since he was Paul’s disciple, etc. There was never just John Layman with no connection with Apostolic office.
It’s true, of course, that *simply *being written by a disciple to an Apostle wasn’t enough to get something considered canonical – it’s a hurdle, not a guarantee. But if it was written by a student and seems to have been based upon their teacher’s teaching and/or recollection of Christ, AND shows signs of inspiration, it was a candidate.
Since the Didache has such unclear origins (moreso than Hebrews, which is clearly Pauline in theme and character, even if not in writing style), it’s not able to clear this bar, even though its teachings are sound.
Obviously, this argument isn’t
de fide, so you’re free to disagree, but I found it persuasive personally when I read it.