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Said correlation is not exclusive. Unless they focused on just respondents who listed watching Fox news, but put "never" down for every other news source (which might drop you to a sample size of one single outlier nutjob in the whole survey), any correlation you attempt to make is going to be meaningless. There's a reason why the better correlations are things like gender, race, how you voted, etc. Those are single exclusive values and you can derive useful information from them while retaining a reasonably useful sample size within the whole set.
Eh, I disagree with that fully. Even assuming that most of the people aren't partial to their primary news source almost to the exclusivity of others (HUGE assumption that I would bet against, and there's probably data to support that, perhaps even cited in the study), no statistically significant correlation is meaningless. It may not mean what you want it to mean, or what you think it means, but it's not meaningless.
Everyone I know who would list FOX as their primary news source really only watches FOX. Everyone else I know refuses to watch FOX except out of morbid curiosity. While television may not be the primary news source of Americans (I honestly don't know) I doubt their news is any less polarized in the websites they visit or the paper they subscribe to. Particularly for conservatives, radio and television are more popular political news sources.
Ultimately it's perfectly defensible to reduce a complex variable like "primary news television station" down to a single binary variable. We do it all the time-- gender, race, and voter affiliation are actually not considered that useful either. In fact, they are increasingly meaningless terms relative to the quantifiable way that they are sometimes treated in social science. That doesn't make them at all useless for determining correlation or causation, but simply draws attention to the fact that these are complex constructs. By comparison, the news source variable is hardly less viable than any other.