I agree with you for the most part. But I invite you to consider why we put so much instinctive faith in NYT/CNN/etc., especially on political topics. What have they done to earn our trust? The fact that they have a neutral tone does not speak to their accuracy. There is big talk about neutralness and impartiality, but do we know that there are effective practices to enforce these things? Have we closely considered the process that produces their content and determined that it maximizes accuracy? I have almost never seen anyone arguing in these terms.
I see no reason to believe that standard journalistic practice leads to high accuracy. Consider that the NYT encourages writers to fit stories to a predetermined narrative:
"By and large, talented reporters scrambled to match stories with what internally was often called “the narrative.” We were occasionally asked to map a narrative for our various beats a year in advance, square the plan with editors, then generate stories that fit the pre-designated line."
http://deadline.com/2016/11/shocked-by-trump-new-york-times-...
Furthermore, consider journalistic practice around quotations--an area where newspapers would want to be extraordinarily accurate, because quotes are easily verified and people expect them to be verbatim. On the contrary, it is standard practice to give paraphrastic, i.e. incorrect, quotations:
http://languagelog.ldc.upenn.edu/nll/?p=6200
None of this is to say NYT is worse than Breitbart, based on my personal judgment. But being better than Breitbart is a pretty low bar. My point is, on what basis do we privilege the old, established media?