The paper isn't the point, the press is. As is evidenced by your dates, linen paper in and of itself does not move the needle much. Asia Minor and China never produced a press. The idea of cheap, disposable paper that leads someone to think up a mass production scheme moves the needle a lot.
The argument put forth in "Paper: Paging Through History" is actually the opposite. Europe was exposed to cheap, disposable paper through trade for centuries but scorned it until they had a need for it.
(While "Paper" is partially a history book, it also has as a thesis that societal change leads to technological change, rather than the opposite, and attempts to demonstrate it through the history of paper.)
I'm interested in reading this. I think I disagree with the main thesis though.
The most important forces in history are geography, demographics, and information flow, probably in that order.
I would propose that the history of the west largely hinges on the printing press. Gutenberg doesn't invent the press, Luther never gets the word out about Reformation, Enlightenment never occurs, John Smith never writes Wealth of Nations, America never splits from Britain, etc.
Note that after the continuous decline in price from 1350 to 1650, the price of paper stabilized due to the demand from printers. The availability of linen rags was another limiting factor until the development of wood pulp paper.
On page 9 -"Vellum and parchment were not viable for mass production. A well-known example is the Gutenberg Bible, of which 30 copies were printed on parchment. Each parchment copy required the skins of more than 300 sheep."
So if you're saying hand-carved woodblock printing should be considered a "press", sure the Chinese had a "press." If you're going to tell me that the printing press could not have been the important factor because a hand-carved woodblock typeset is just as economical as an actual press with metal movable type, I'm willing to debate you on that.
They had mechanical printing presses used for wood-blick illustrations. They also had movable type printing presses for long text. For some reason they didn’t combine the two until much later. Probably market forces though—Europe was rather unique in having demand for everyone to own a Bible during the reformation. Without needing thousands of copies, hand presses were easier.