An important point, I think: The propagandists, most prominently today the News Corp outlets such as Fox and the WSJ, want to portray the whole world as ideologically driven. It innoculates their followers against anything anyone else can say (e.g., 'it's all left-wing propaganda, don't even bother') and justifies their own propaganda.
But everyone is not ideological. MSNBC is, at least it was the little I saw of them awhile ago, maybe NPR is, but I disagree that the rest are on the left. They are mostly straight-ahead journalism without significant ideology (though of course nothing is perfect in the world, and I'm omitting the unconcious ideology of the US popular perspective that people like Chomsky observe.)
I doubt it's possible for the news to be truly non-ideological for this, and many other, reasons. The best one can hope for is a news organization that is a) straightforward and open about its political viewpoint and b) reports in a sober and accurate manner with a minimum of flinging partisan red meat around. There are a fair number of American news sources that are a), but b) is much harder to come by, especially on television. Given this, I can understand why some Americans flock to the BBC or Al Jazeera or whoever, even if I personally despise their editorial positions.
It's more straightforward than you might think, you just have to give airtime to conflicting views, and make sure discussion of news stories are conducted respectfully.
I understand what you're saying about it being possible to show bias just by what news stories you choose to cover, but inviting on dissenting voices helps tackle that issue. If you expect intelligent debate from your news sources, you'll spot it when it's absent.
To put it another way, imagine the Guardian vs. the Daily Mail, but on television instead. No one, I hope, would pick up a copy of those newspapers expecting the unbiased facts; same deal here.