Was there really no Bible until the year AD381?

I'm writing on an urgent matter that I think needs to be addressed within Catholic apologetics, namely the widespread Catholic claim that there was no Bible until Pope Damasus gave us the canon of Scripture in AD381. At best, this is a half truth, and at worst this is an implicit heresy and undermines Christianity. While it might score points against Protestants during a Sola Scriptura discussion, it's a bad argument that does far more harm than good. 

The problem with the "no Bible until 381" claim is that those who make the claim typically have in their mind that in the early Church - sometime after the Apostles died (~AD80) and up to 300 years later (~AD380) - there was mass confusion as to what books were Scripture and what weren't, such that the Pope had to call a Council to settle the matter by sifting through a massive pile of books, some which were inspired and some which were uninspired, and the "result" was the canon of Scripture. This mindset suggests that the Bible wasn't something passed onto us by Tradition, but rather something that was basically invented. The Pope most certainly did not walk into a library and start reading random books and try to "detect" if this or that book should be in the Bible.  In fact, this false 'personally feel out if this book is inspired' method is closer to the Protestant and Mormon approach to the canon of Scripture.

The true history is more along the lines of this: Even as the Apostles themselves were writing the New Testament, the NT writings were preserved and circulated to major congregations by trustworthy Christians (Col 4:16). While we aren't sure how widely each NT writing was circulated, the earliest of the Church Fathers quote from basically every NT book, meaning all 27 books of the New Testament were mostly gathered into mostly complete collections within at least two generations, i.e. by around AD150. There never were various non-inspired books lingering around the NT writings, as if the early Christians were confused about whether a writing is Scripture or not. There were some early Christian writings that were held in very high regard by the earliest Church Fathers, such as the Epistle of Pope Clement in AD90, but they never considered these as Apostolic, Biblical writings. 

There were a few Councils and Fathers who had questions on what status certain books held, such as the Deuterocanonical Books, because some churches had them in their collections of Scripture, while others did not, but this is very different from presenting a situation where there was a pile of random books that the Pope and Bishops had to sift through. Augustine said that the rule when it came to books that were considered Scripture by some but not by others was to look towards the more ancient congregations, which is where Augustine came to accept the Deuterocanonical Books right along with the Protocanonical Books. It was then in this context that the Pope and Councils would later come to speak more definitively on a few books that were disputed, such as to help get all churches on the same page.

Approaching the canon from this more mature and historical perspective still leaves the Protestants in a bind as to how to where we got the canon of Scripture, since this did come from Tradition telling us what books were inspired, while it also preserves the (true) historical fact that the Apostles really wrote the Bible and these writings were faithfully passed on, along with the Liturgy.