Two things to keep in mind. First, the book buying market was probably bigger when Gutenberg did his thing. Second, printing is most useful for books which are in high demand, which for the classical world means dream interpretation manuals and omen handbooks.
I think this POD will make a difference, and progressively a greater difference, but lets not get carried away.
Gutenberg actually wasn't taregeting the book market. His business was formulaic documents that every scribe hated making out, but that needed to be produced in their hundreds or thousands. Printing the Bible was more a proof-of-concept idea (we
can make books), and financiallky ruinous despite the fact that he was able to undercut prices for handwritten books.
The market for books in Classical Athens was probably much, much smaller than it was in 15th century Germany, but by the end of the classical and the Hellenistic era, it would have grown enormously. There were, by then, businesses that had scribes copy the most popular works in the expectation of ready sales. Hellenistic Athens could have used printing very well to export its culture yet more effectively.
Unfortunately, the limits of technology make it very improbable. Not because literate slaves would be cheaper than printing - both buying and training a scribe would cost serious money, and you could not starve a slave of such quality. The cost of employing skilled slaves or freemen was equal for third parties, and the income generated by slaves for their owners was still not high enough to make the model attractive. The problem is that there are going to be hundreds of details that need to be gotten right, and without a lot of previous work to build on, it is hard to see how they would get there.