I am no expert, but let me take a stab at this according to my own admittedly limited knowledge and understanding.
I think that the key is that the Bible is looked at in two different ways: In one sense, as a collection of historical writings and letters, and in the other as Divinely Inspired Scripture. For reasons below, it is legitimate to use these two different understandings almost as if they were in fact two different sources rather than one circular source. At this point that sounds weird, so read on to understand why it works…
We do not first start with the Bible as inerrant. We start with the Bible as one piece of
historical evidence among multiple indicators that Jesus Christ was a real human being, Who really claimed the things we believe He claimed, and Whose disciples and followers claimed that He rose from the dead under highly unique circumstances in which their lying or being deceived was highly unlikely. And indeed, one need not even look into the Bible to find that there were people who claimed that Jesus had risen from the dead. Once we realize that Christians really did exist at the time of Jesus’ death, we can turn to the Biblical epistles, not as scripture but as mere letters, to confirm what the early Christians were
saying about Jesus: That there were eyewitnesses to his death, and that these eyewitnesses themselves were willing to die for those claims. We can believe these epistles’ claims about this not because “The Church says so” but because if these claims had been pulled out of thin air, the fact that these letters were written during the lifespan of Jesus’ contemporaries would mean that they would have been easily shot down and the Christian movement would have been over before it began: After all, Jesus’ life was highly public and debunking obvious myths about Him would have been simple. So we know, from using the Bible in strictly historical means, that the earliest Christians believed that Jesus died, rose from the dead, and that the Church was the “pillar and bulwark of truth” because these are all contained in those portion of scriptures that were letters, and no substantial number among the early Christians seems to have disputed this.
As for why we can conclude that the earliest Christians were
right about Jesus’ resurrection, that is a different discussion altogether, but to touch on it lightly here it’s because of the unique circumstances surrounding the claims. If Jesus didn’t rise from the dead and ascend to Heaven, then either Jesus managed to pull off a hoax that no one since then has ever been able to pull off with similarly limited technology (not only a resurrection, but being taken up into Heaven before the eyes of witnesses), or all the eyewitnesses were crazy in the exact same way (they were willing to die for what they claimed to see with their own eyes), or they were all crazy enough to die for what they absolutely knew to be a lie. To me personally, the Christian claim seems the most reasonable (yes, atheists and other unbelievers will balk at that, but I’m just being honest

). And if Jesus did rise from the dead, then the early Christians’ beliefs, including that about the Church (which they said was “the pillar and bulwark of truth”), is something to be taken more seriously. Even aside from that, we know simply from quotes from the early Church Fathers that the Church was considered to have authority and, as an entity, to be incapable of erring from Truth.
And once we have concluded that the Church has Divine teaching authority–that is to say that the Church is the “pillar and bulwark of truth”–we can from there confirm that the
rest of the Bible is accurate (in the sense of “inspired”, not always in the sense of being taken literally), even those parts which are very difficult to verify from history (such as parts of the Old Testament, such as the book of Revelation, etc.) are accurate, because this is what the Church teaches.
So rather than saying “The Bible is accurate because the Church says so, and the Church is right because the Bible says so,” there is a subtle difference. We are actually saying “Certain parts of the Bible are most likely accurate because their being accurate makes the most sense in terms of the other evidence from history and in terms of logic surrounding the historical context; those parts of the Bible coupled with quotes from relatively early Christian leaders lend credence to belief that the Church is the source of truth; and the Church teaches that the
entire Bible is inspired, by having canonized it, rather than merely
happening to be accurate.”
The distinction may seem subtle, but it’s there, and it makes all the difference in the world.
As an aside, when you think about that last part, it is only if there is an authoritative and infallible Church that we
can conclude that the Bible (at least the New Testament) is anything more than a collection of writings that “happen” to be accurate, because the New Testament, at least, was never claimed within its own writings to be written as Divinely Inspired Holy Scripture with the possible exception of the book of Revelation.
I hope this helps!
Blessings in Christ,
KindredSoul