Whenever anyone challenges me about the church's role in choosing the canon I still come back to a quotation I read in my first few years of being a Christian:
"the church did not select and authorize the canon but recognized it and sat down under its authority. When a patient returns to a doctor and says, 'Yes, that medicine is exactly what I need', the patient does not authorize the medicine but recognizes its inherent authorization." (p. 207, Alec Motyer, Look to the Rock)
One reason why I think Motyer's point works, even if it could be refined, is because it does not separate the Bible's authority from its purpose. You cannot know the authority of scripture apart from its saving work in your life, and the early church didn't try.
But how did the early church know that the purpose of any books claiming to be the word of God should be salvation of sinners? They could, and to some extent did, think of it as a political tool, a guidebook to life etc instead. However, from creation onwards the church has experienced the word of God and has known that it is characterised by the creation of life. The word of God and God's saving activity have always been inseparable.
So when it came to deciding the canon the primary question asked of any book was does this book bring salvation? Questions about historical reliability and apostolic authority are also valid, but are correctly understood as part of this question. The church cares that they provide historically reliable access to Jesus of Nazareth because he is our Saviour, and we care that their authority derives from Jesus because only he has the authority to forgive our sins.
But how do we know if a book brings salvation? We know salvation by the Spirit who gives life and freedom (John 6:63; 2 Corinthians 3:17). In this sense the Spirit authenticates the Bible. Not by a little voice in the head saying "yes that's true" or "no that's a fake", but by the new life that accompanies it (cf. John 3:8). Paul appeals to the Galatians experience of the Spirit ("Did you receive the Spirit by works of the law or by hearing with faith?", 3:2) not just to verify the sufficiency of faith but to verify the Gospel of Christ he preached and they heard (1:6-7). God's word, God's salvation and God's Spirit are all inseparable. Our salvation is hidden with Christ, but we have the down-payment of the Spirit, so if we have the Spirit we can know we have life as an inheritance waiting for us (cf. 1 Peter 1:4-5).
But finally, how do we know if we have the Spirit? Paul and John both agree on one answer to this:
"No one can say 'Jesus is Lord' except in the Holy Spirit" (1 Corinthians 12:3)
"By this you know the Spirit of God: every spirit that confesses that Jesus Christ has come in the flesh is from God" (1 John 4:2)
So to a degree, we can ask the question about the canon apart from our personal experience. We can ask, does the Bible confess Jesus Christ as Lord? Jesus thought the OT did (John 5:39) and I'm convinced all the NT does too. And if it does, when we can be sure that it is the Spirit-breathed, salvific word of God. But when I have come across non-canonical 'Gospels' I've been less convinced they confess the Jesus as Lord.
In Matthew and Luke I find that Jesus is confessed as the Lord who goes searching after lost sinners like a good shepherd (Matthew 18:12-14; Luke 15:3-7). In the Gospel of Thomas I find a Jesus confessed as the Lord who who also goes searching for a lost sheep, but that sheep also happens to be the largest sheep and is never identified with sinners (saying 107). That seems like a different Jesus, and a different Lord to me, and it brings no salvation for a lost sinner.