In Eastern Mediterranea, literacy rate remained high until at least 6th century (if I recall correctly. CMIIW though). Bibles were translated to vernacular language, like the Syrian bible Peshitta. Book copies were plenty, as suggested by the fact that destruction of Library of Alexandria didn't actually set us back for hundreds of years.
A situation very different from what we usually know in middle age Europe. The return of rising literacy rate, Bible translation, and books in vernacular language in 16th century Europe then is highlighted as an important development that led to modern nation, such as indicated by Benedict Anderson.
What makes such vernacularism different from the one we had in 2nd-4th century? Did the 2nd-4th century vernacularism have similar impact like the one in 16th century?
I don't know an excessive amount about the vernacular of the 2nd to 4th centuries, but I can tell you one thing that absolutely made a huge difference: the printing press. This, along with early and complete political protection for Luther, is often cited as one of the two most important things that allowed the Protestant Reformation to thrive, instead of being crushed like other Reform movements in the Latin Church (Waldensians, Jan Huss, Wycliffe, etc.).
Benedict Anderson famously argues that "print-capitalism" led to nationalism. That is, people wanted to read stuff in the vernacular. So they bought stuff in the vernacular. So printers printed more stuff in the vernacular for this community of readers (this is the capitalism part--supply meeting demand--which was impossible without modern modes of mass production). This created an "imagined community" of readers reading in the same language, which ultimately gave rise to the nation state. Now, Anderson is wrong about this, at least partially (his early case, Latin America, most of what was being printed was either royalist governmental or religious, hardly the stuff of an imagined national community), but it has been very influential in the field.
The thing is, if you look at ~16th century, you realize how much more stuff is being printed everywhere. Not just books, but pamphlets, woodcut illustrations, etc. There are traveling book salesmen and this spreads ideas much faster, both across territories and across classes. I think, though, more than anything this chart explains what was different in the 16th century from any time before it. See also here, for a more direct comparison of manuscripts and printed books but pay attention to the logarithmic scale! For example, in the 15th century it looks like manuscripts and printed books are close, but in reality, there are more than twice as many printed books already (12,589,000 vs. 4,999,161). The change from the 6th century to the 16th is several orders of magnitude (13,552 up to 217,444,000). Thats well over one and a half million percent growth!
So what I'm saying is there were plenty of books that one copy being destroyed didn't set back civilization, and there's plenty of books that millers and farmers and the like could actually own them. In Carlo Ginzburg's famous The Cheese and the Worms, we find out that a weird but not at all wealthy Italian miller owned several books. We don't know exactly which ones he owned (because none were banned so no records were kept) but we have a list of at least eleven books that he was known to have read (we don't know how many were borrowed vs. owned). But the point is, many more books were in much wider circulation in the 16th century compared to the 4th, whether you buy Benedict Anderson's argument or not.