Woodblock printing was invented in the 7th century in Tang China.

The prerequisite for economic printing is paper, and the tech tree for paper is reproducible from Roman conditions. The production of vellum is measured in years, scribes were barely the limiting factor on text production.

There is a lot of basic scientific knowledge that many/most modern people have (germ theory of disease, how the body works at some level, astronomy 101, etc.) But ancient people weren't stupid. They just didn't have advanced technology. Recreating a whole chain of technology to bring about the iron age maybe 1000 years early probably isn't happening. The fact is that they're mostly doing pretty well with building things that the technology more or less exists for.

I also read somewhere that a good deal of his innovation was also in finding inks suitable for the process.

A lot of existing inks at the time where suitable for writing not pressing.