It's fun to point and mock and I certainly engage in it myself, but most people don't pay attention to politics very much (nor should they have to), and even those who do aren't exactly well-informed by their sources, whether we're talking about Fox, CNN, or NPR. Sure a lot of the nonsense comes from the explicit right wing news, like Fox, but the bothsidesing of even "objective" news doesn't leave a lot of room for certain truths to be reported as fact.
We're a big racist country filled with racists, and that's why people are so quick to believe in things like "Obamaphones" and the existence of the secret welfare system for black people generally, but they don't just come to those ideas on their own. It's what they've been hearing about, even from supposedly respectable news outlets (and while Fox's rep fell during the Obama years, other news outlets treated it with respect for a long time), for a long time. Those conservative think tankers dress up the racism with some fancy words before they go on NPR, but they still go on.
I get annoyed be the "people vote against their self-interest because they're racist!" idea because I don't actually believe it's true. I think people are racist and it makes them believe stupid shit which then gets them to vote against their self-interest, but they don't think they are doing that. They're racist assholes for wanting to take away health care for black people, and they're mostly idiots for thinking that their health care won't be taken away, too (though "single payer for white people" seems to be the preferred conservative policy solution these days), but they aren't just voting to stick it to black people. They think black people are getting all the good stuff and they're mad that they don't get any of it. This is false, of course, and only something you could believe if you're pretty damn racist, but the "stupidity" is understandable even if the racism isn't forgivable (which it isn't).