I've been reading John K. Fairbank's China: A New History (in French, so I can only paraphrase instead of quoting directly). He indirectly touches on a reason for that near the end of chapter 8 of the book, about the economy in the Qing dynasty. Here's a rough summary of what he writes: the concept of a corporation with a legal personhood separate from its owner(s) didn't exist. Any large business was a family business. Commercial relations were one facet of social relations and so were influenced by things like friendships and family obligations instead of being governed by contracts.

So (I'm no longer repeating what Fairbank wrote now) to extrapolate, all this would have come together to make it very difficult to invest into businesses (such as newspapers) beyond the short term. It's a similar reason for the fact that newspapers in Europe start appearing around 1600, about 150 years after the printing press appeared. Legal concepts like corporate personhood were of course introduced in China in the 19th century, and that's when newspapers start to appear, as you noted. Mind you, this takes all of one paragraph in Fairbank's book. There's probably some other work with more precise information, but it does provide the beginning of an explanation.

Also, I can't help but notice that the "Media history of China" article is only talking about computers. That's why it starts in 1956.