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Republicans and Democrats differ drastically in which news sources they trust and distrust
(www.pewresearch.org)
Source: Pew Research
I think it's less about having an agenda or not, and more about the factuality of reporting. If I read "so-and-so is quoted as saying..." then can I trust that what is being represented is the same as what the speaker intended?
Agenda-free reporting would be nice in theory, but I don't think it's very realistic. We can demand certain standards of factuality though, and then just do the legwork to identify and take biases into account.
I mean, they are nowhere near Rogan or Fox News in terms of agenda, but i think Rogan is a great example. Most of his stuff is just podcasts. So instead of "quoting" the guests, he just lets them speak directly. But the choice of guests, the choice of question and so on, make a huge difference in the picture conveyed to the consumer.
One could argue that the relatively subtle way of media like the NYT is more insidious as it is harder to spot, but it seems that the "blunt" approach of Fox News isn't less effective, so i couldn't judge which is better or worse.
I agree with you that it isn't very realistic to find reporting free of any Agenda, even if the reporters do their best to be as neutral as possible.