You're correct that I put more faith in actual news networks than in overtly partisan blogs, but your reasoning is completely wrong. Did you read that list of things I do to check the validity of a story?
Let's taker the NYT as an example. You'll probably tell me that it's an untrustworthy, biased, liberal rag. Here's the thing, though - stories in the NYT cite sources (usually) and try to connect the events being reported to the bigger picture. If the Times reports something I can usually verify their reporting, either immediately or soon, by cross referencing their claims with news from other outlets. Beyond that, if the Times screws the pooch and gets a story totally wrong they'll post a correction. And when they're just doing opinion pieces they clearly identify them as editorials. Real news organizations have journalistic standards. They're not perfect and sometimes they make mistakes, but they do their level best to find and report verified and/or verifiable facts.
My point here is that, no, I don't automatically assume that a source like the Times, the Washington Post, the WSJ, or the Associated Press is accurate and they don't ask me to make that assumption. They make evidence-based claims and cite their sources so that I can go look for myself and form my own conclusions.
By comparison, the hyper-partisan punditry blogs you're often citing here make no effort whatsoever to verify if a story is true. They don't try because they don't care. Their job is to push a narrative, so they seek out incidents that support that narrative. If things don't quite line up they fudge the details, ignore the context, or hide the boring truth behind a wildly sensationalized headline. They don't do news, they do spin.
The difference isn't that one source leans liberal and the other is conservative. The difference is that one source does journalism and the other does punditry. Journalists support their own claims. Pundits don't, so their claims should be viewed with skepticism and independently verified when possible.