In part, Rome’s decision was an attempt to cut off the traction being made by the fledgling Reformation.
The New Testament Canon is mainly written by the apostles. Grudem writes:
“It is primarily the apostles who are given the ability from the Holy Spirit to recall accurately the words and deeds of Jesus and to interpret them rightly for subsequent generations.”5
Apostles in the early church have claim to the same authority as the Old Testament prophets and are able to claim they are speaking (writing) the very words that come from God.
So, what of the other New Testament writers of Mark, Luke, Acts, Hebrews, and Jude? Hebrews aside the writers of these books had a close assocation with the apostles. The were able to write what they did based on first-hand accounts. Hebrews was a challenge for canonicity because of the questions surrounding its authorship. Many scholars say that it was written by Paul, and there is some evidence of Pauline authorship. In the end, it came down to the tesitmony of the book itself. According to Grudem, “Thus, the acceptance of of Hebrews as canonical was not entirely due to belief in Pauline authorship. Rather, the intrinsic qualities of the book itself must have finally convinced early readers, as they continue to convince believers today, that whoever its human author may have been, its ultimate author can only have been God himself.”6
When it comes to books like the Shepherd of Hermes or the Gospel of Thomas, these writings were so obviously flawed that no serious scholar considered them worthy of canonicty.
As I alluded to in your OP, this can become a protracted discussion with no accepted conclusion to all involved. But speaking from a decidely Protestant and Baptist perspective, the 66 books of the the Bible that were attested to by the Reformers, Puritans, and Baptists today is the the very word of God, and is “the only sufficient, certain, and infallible rule of all saving knowledge, faith, and obedience.”7
1. Grudem, Systematic Theology, page 56
2. Ibid., page 57
3. Ibid., pages 57-58
4. Ibid., page 59
5. Ibid., 60
6. Ibid., 62
7. 1689 Second London Baptist Confession of Faith, Chapter 1.1