Where does our modern doctrine of Biblical inspiration come from?

Why do we believe that the Bible is inspired? Where does that belief come from?

Now, look, I do know the basic answer. 2 Timothy 3:16, "All scripture is inspired by God and is useful for teaching..."

However, I think there's a couple issues with this:

1. This is circular logic. The Bible is inspired because the Bible says so. However, the Qu'ran also claims to be inspired. Do we believe that?

2. There are some serious scholarly debates over whether "inspired" is actually the correct translation of this passage in 2 Timothy.

3. Paul (or, to be more accurate, the person pretending to be Paul) was likely referring to the Hebrew Bible. At the time of Paul's writing, there was no New Testament canon yet. If you were to accept the traditional authorship of this letter, Paul died before the gospels were even written, so "Paul" certainly wasn't referring to those.

So, with all this in mind, how (and why) do we know that the current canon of the Bible is inspired by God? The books of the New Testament were selected by the early church fathers, largely by consensus. How can we be sure that they were 100% correct in terms of what books they selected? There have, after all, been disputes over time. Protestants reject the Deuterocanonical books (in the Old Testament), and Martin Luther disputed the legitimacy of Revelation, Jude, and a few other books.

What if the early church fathers made some errors? What if, say, The Shephard of Hermas, or the Didache, really were inspired? What if James wasn't?

In fact, there is incredibly compelling evidence that the church fathers DID make mistakes. Take for instance, 2 Peter, which scholars almost unanimously agree was not written by Peter. Or take the letter to the Hebrew: this book was included because it was believed to have been written by Paul, even though it does not even claim to be so.

Now, I also know that a common argument against this is that pseudepigrapha (that is, pretending to be a famous author when you are not) was a common practice at the time, and was not thought to be dishonest. This is, according to the scholars who actually study this topic, really not true at all. Even in the Bible, there are warnings against letters impersonating Paul: see 2 Thessalonians 2:2. Furthermore, letters known to be forged, like Paul's Third Letter to the Corinthians, were not accepted into the canon.

I understand the basis of belief for something like the resurrection — it is based on the testimony of the early apostles, and those who wrote about them. But I'm confused about Biblical inspiration. Christians seem to treat it as self-evident, but it really isn't evident to me at all.

So, with all that said, where does the belief in the inspiration of the Biblical text come from? Is it correct? Is it reasonable? Does it hold up under scrutiny?