Can we see the paper problem being the other way round.
No we cannot.
As the Phaistos Disc shows, printing has been around since at least the second millennium BCE. Roller seals, signet rings, woodcuts, and all the rest have been used to print on various substances for nearly as long.
What far too many people completely fail to realize it that the printing press simply allows for
mass printing and that
mass printing requires a
mass printing medium.
Advances in producing a printing medium, i.e. paper, meant that the price of books could drop substantially
IF you could only print books quickly enough. The materials were there and the market was there, but the existing production methods meant you just couldn't produce books fast enough or cheaply enough to meet demand.
Typographic woodblock printing was already producing books but carving a separate woodcut for each page was incredibly time consuming, something on the order of weeks, and pretty damn expensive. Movable type cut that setup time to a few hours and the hand mould meant you could produce your own type instead of hiring an artisan to carve a woodblock. While a Gutenberg press could "only" print twice the pages per day a woodblock press did, you could create a page for a Gutenberg press in a fraction of the time a page for woodblock press required and that page would be a helluva lot cheaper.
Of course, this need for speed wouldn't exist if large quantities of relatively cheap paper weren't available so, no, we cannot see the paper "problem" the other way around.