Under its definition for paper, Macmillan says:
paper (n.) a newspaper
He sat down and read the paper.
The story was in all the papers.
That first example sentence is common parlance – we often say "the paper" to mean "a newspaper" or "today's newspaper".
We could paraphrase the original as:
Their major voters are dependent upon television – they don't get their news by reading the paper or by using social media.
The original sentence does seem a little awkward to me, but not because it doesn't say "reading papers". I think "reading the paper" works here because the author is referring to a habitual action. It seems a little awkward, though, because I'm unaccustomed to hearing, "reading social media," even though I am very used to hearing, "reading the paper."
If I wanted to know if you read through a newspaper on a regular basis (not necessary daily, but even just a few times a week), I would be more likely to ask you, "Do you read the paper?" instead of "Do you read papers?" The second question sounds more like I'm asking about academic journal articles instead of the daily print news. When talking about newspapers, I might even ask, "How often do you read the paper?" but not, "How often do you read papers?" ("How often do you read newspapers?" is better, but I still prefer "...do you read the paper?")
When reading your quote, I think it's important to understand that "reading the paper" is very idiomatic, at least in American English.
Note the rise in usage of that expression in American English over the past several decades on Google ngram. The British English corpus doesn't show the same trend.