I think it's possible to do, just takes time.
With non-Western affairs, I try this but feel like I still struggle with obscured facts and deep-seated agendas. The errors and biases in multiple accounts of a domestic issue can often be subtracted out, but English coverage of complex foreign issues has bigger complications. Distinct authors work from the same erroneous primary source, or all make the same cultural misunderstanding on the same point, or all lack key context that wasn't international news.
Coverage of topics like the war in Syria is often mind-bendingly bad, and I end up running down primary sources even when I can trust them, just for context. (A concrete example: many stories about refugees in Raqqa listed populations and areas that would have left the surviving city more densely populated than Kowloon, but no one seemed to bother doing that division.)
It's vastly better than it used to be, though. Total inaccuracy is much less of a threat in a world pervaded by English-speaking internet users. I feel surrounded by sloppy, unhelpful media, but I at least have enough information to get that feeling. Stories like the Western media's misunderstanding of the USSR in the 1930s give some perspective on what it looks like to be so comprehensively wrong that you can't even tell there's a problem.
It helps filtering for intentional spin, but not much with reporters just being flat out wrong.
I've got half a dozen tech news sources I kind of trust (as in, trust that I can usually spot their errors) - Ars, Techdirt, and their ilk are all at least useful. WaPo/NYT/etc are largely subject-specific to US politics, but in denial about it, so that's workable. Physics and biology news I'm largely reduced to reading blogs and asking friends.
I've found ways of making this work for lots of things, but medicine and foreign affairs are the two that give me fits.
Medicine, like social psychology, is full of headlines reading "X causes Y!" over studies finding "X correlates with Y at p=0.30". I end up just reading the damn studies myself, which is obviously not efficient or trustworthy.
And foreign affairs is basically hopeless. I'm easily 5 removes away from facts at when reading an English-language story about, say, Saudi domestic politics. There's effectively no 'ground truth' to access, because every source I can compare to has its own issues. It's honestly sort of baffling when I hear people talk about those topics like they're fully-informed; I'm not sure how anyone who isn't an actual expert in the specific foreign affair in question works up that kind of confidence.