The answer to your question is that some in the early Church did and some didn’t.
Eusebius (c.260-337 AD)
Ecclesiastical History, 3.25.1-7.
It is reasonable to sum up here the writings of the New Testament that have been mentioned. First place must be given to the holy tetrad of the Gospels, followed by the Acts of the Apostles. (2) Next one must list the epistles of Paul; after them the so-called first epistle of John must be recognized, as well as the epistle of Peter. Following these, if it should seem appropriate, one must put the Revelation of John; we shall set forth opinions about this book in due time. (3) These then belong to the accepted writings.
Among the disputed writings that are, nevertheless, familiar to the majority there is extant the epistle said to be by James, that of Jude, the second epistle of Peter and the so-called second and third epistles of John, whether these are by the evangelist or by someone else with the same name. (4) Among the spurious writings must also be counted the Acts of Paul, the so-called Shepherd, the Revelation of Peter, and, in addition, the so-called Epistle of Barnabas and the work known as the Teachings of the Apostles and, moreover, as I said, the Revelation of John, if this should seem to be the right place for it. For some, as I said, reject it as spurious, while others reckon it among the accepted writings. (5) Now some have also counted among these writings the Gospel according to the Hebrews, which is especially favored by the Hebrews who have accepted Christ. (6) These would all be among the disputed writings, but we have been obliged to make a list of them also, thereby distinguishing the writings that according to the ecclesiastical tradition are true, genuine, and accepted and those that, in contrast to these, are not [encovenanted] but disputed and yet known to most people in the church.
We have done this in order that we might be able to know these writings as well as those published by the heretics under the names of the apostles, on the pretense that they comprise the gospels of Peter and Thomas and Mathias or even some others besides these, or the Acts of Andrew and of John and of the other apostles. Not a single one of the ecclesiastical writers through the years has thought any of these worth mentioning. (7) Moreover, the type of language is different from apostolic usage and the thought and purpose of these writings is totally at odds with true orthodoxy, showing clearly that they are the work of heretics. For this reason, then, these must not even be classified among the spurious writings but instead shunned as totally absurd and ungodly.
Eusebius makes three categories for texts,
Accepted/Acknowledged
the holy tetrad of the Gospels, the Acts of the Apostles, the epistles of Paul, first epistle of John, the epistle of Peter, the Revelation of John
Disputed/Spurious
James, Jude, the second epistle of Peter, second and third epistles of John, the Acts of Paul, Shepherd, the Revelation of Peter, Epistle of Barnabas, the Teachings of the Apostles, the Revelation of John
Absurd/Ungodly
the gospels of Peter and Thomas and Mathias or even some others besides these, the Acts of Andrew and of John and of the other apostles
It seems that in Eusebius’s list there were two basic ways the books were evaluated, (i) genuine authorship, and (ii) orthodox teaching. Eusebius puts books into his second category of “disputed/spurious” when they are of dubious provenance but are thought to contain orthodox teaching. It may also be the case that an additional factor is that Eusebius found evidence of their use in churches.
The way Eusebius tried to construct his list was based on his historical analysis of the writings of Christians.
“according to the ecclesiastical tradition are true, genuine, and accepted and those that, in contrast to these, are not [encovenanted] but disputed and yet known to most people in the church.”
“Not a single one of the ecclesiastical writers through the years has thought any of these worth mentioning.”
For analysis of Eusebius see Everett R. Kalin, “The New Testament Canon of Eusebius,” in The Canon Debate , eds. Lee Martin McDonald and James A. Sanders
As Bruce Metzger notes in The Canon of the New Testament: Origins, Development, and Significance, Eusebius was in a position to be able to access texts produced by Christians
“The ‘Father of church history’ had at his disposal the library at Caesarea which Origen built up… He had read a prodigious number of authors, and in the extracts that he gives from their writings he never fails to note the use they made of Scripture, the list of books they quote in passing or fully discuss, the judgments they pronounce on them.” (pp.201-202)
Analysis of manuscript data of early Christian texts by Larry Hurtado in The Earliest Christian Artifacts: Manuscripts and Christian Origins gives a similar picture.
Also using Biblindex (http://www.biblindex.mom.fr/) as a rough estimate of allusions and quotations up to 325 AD correlates well with Eusebius and the manuscript data.
Augustine (354-430 AD) and Jerome (347-420 AD)
When we get to Augustine and Jerome we see a different approach to canonicity,
On Christian Doctrine, 2.12
Accordingly, among the canonical Scriptures he will judge according to the following standard: to prefer those that are received by all the catholic churches to those which some do not receive. Among those, again, which are not received by all, he will prefer such as have the sanction of the greater number and those of greater authority, to such as are held by the smaller number and those of less authority. If, however, he shall find that some books are held by the greater number of churches, and others by the churches of greater authority (though this is not a very likely thing to happen), I think that in such a case the authority on the two sides is to be looked upon as equal.
Augustine doesn’t consider authorship to be a reason to receive or reject a text.
Jerome says this about the author of Hebrews,
Letter 129
“It does not matter who is the author of the Epistle to the Hebrews, for in any case it is the work of a church-writer and is constantly read in the churches”