I think the rhetorical term is apophasis. It's the same sort of deniable invocation as declaring "I won't stoop to talking about my opponent's sordid affair". A rhetoric teacher might say that it's just another way to persuasively make a point, but it seems hard to deny some manipulation or dishonesty when the "I won't say" element is intentionally false.
You make an interesting point about the recent/online deployment of it. Most examples of apophasis are about bringing up familiar topics without owning that action, as with invoking rumors or ad hominems in a debate. But I increasingly see the version you're describe, where it's used as a way to present a very specific statement without needing to defend it. A lot of old uses of the technique only work as a full sentence, specifically to avoid being quoted as saying the inappropriate thing directly. But the new ones do the opposite, very specifically accommodating misleading pull quotes. "As a recent Guardian article about critical thinking courses said, 'Fox News promotes "fake news"...'" The phrase comes into existence, soundbite ready, without anyone needing to actually defend it.
Perhaps this isn't the most extreme example, since the author might readily defend the "fake news" claim, but it still serves to associate the college course with rejecting Fox when that's expressly not the point.