Not many direct statements about the lack of canonization of those books were made during the specified time. But a few statements were made.
The canon list in the Muratorian Fragment, dating to about 170, mentions the Shepherd of Hermas, saying that it "cannot be read publicly to the people in church" because "it is after [the apostles'] time":
Hermas wrote The Shepherd very recently, in our times, in the city of Rome, while bishop Pius, his brother, was occupying the chair of the church of the city of Rome. And therefore it ought indeed to be read; but it cannot be read publicly to the people in church either among the Prophets, whose number is complete, nor among the Apostles, for it is after their time.
That's the only statement I could find that mentions a specific book of the Apostolic Fathers and gives a specific reason for its exclusion from the canon. But I'm not going to end the answer there.
Eusebius, the fourth-century church historian, has four categories of books:
"recognized," which seems to mean that there is a consensus that they belong in the canon
"disputed," which seems to mean that a majority recognize them but some dispute their canonical status
"spurious," which seems to mean that most don't count them as canonical
"heretical," which means that nobody in the church has ever regarded them as trustworthy
About the heretical books he says:
To none of these has any who belonged to the succession of ecclesiastical writers ever thought it right to refer in his writings. Moreover, the character of the style also is far removed from apostolic usage, and the thought and purport of their contents are completely out of harmony with true orthodoxy and clearly show themselves that they are the forgeries of heretics. For this reason they ought not to be reckoned among the spurious books, but are to be cast aside as altogether absurd and impious.
From this we can infer that Eusebius' criteria for considering a book canonical is that it 1) is accepted by the church, 2) is orthodox, and 3) is from the time of the apostles. He regards Barnabas, the Didache, and the Shepherd of Hermas as "spurious," but doesn't mention any other of the Apostolic Fathers.
Athanasius seems to have a threefold categorization:
Canonical. "These are fountains of salvation, that they who thirst may be satisfied with the living words they contain. In these alone is proclaimed the doctrine of godliness. Let no man add to these, neither let him take ought from these."
Good non-canonical books. "Not indeed included in the Canon, but appointed by the Fathers to be read by those who newly join us, and who wish for instruction in the word of godliness."
Apocrypha. "They are an invention of heretics, who write them when they choose, bestowing upon them their approbation, and assigning to them a date, that so, using them as ancient writings, they may find occasion to lead astray the simple."
Of the Apostolic Fathers, Athanasius only mentions the Didache and the Shepherd of Hermas, both of which he places in the middle group.
Rufinus says that "the ancients" "were willing to have all these read in the churches but not brought forward for the confirmation of doctrine," referring to a handful of books including the Didache and the Shepherd of Hermas, though he doesn't mention any other Apostolic Fathers.
The word they were all using for canon can mean "rule" or "standard" or "measuring stick."
So, although the direct evidence is relatively scant, we can infer the following from the aforementioned facts:
Anything not from the apostles' time was disqualified (which disqualifies most of the Apostolic Fathers, although some of them, like 1 Clement and the Didache, may have been written while John and others were still alive.)
Anything that misrepresented its authorship was out (which probably only disqualifies the Epistle of Barnabas, 2 Clement, and the inauthentic Ignatian epistles.)
The canon was to be the final appealed-to doctrinal word (Athanasius and Rufinus are the clearest expositors of this point, but the word "canon" itself seems to be a clue to this fact.) Jesus, Paul, and John are probably the clearest New Testament examples of speakers/writers who claim to be speaking directly from divine revelation. Perhaps 1 Clement and the Didache, which I mentioned in point 1 as possibly from the apostles' lifetime, were excluded for consciously being doctrinally dependent on the apostolic writings. But that's speculation.
Consensus. Eusebius and Rufinus both seem to believe that the most important canonical criterion is whether "the church" through the ages has accepted the book as fitting the other three criteria. Perhaps the councils were able to make their choices because the consensus had already solidified by that time.