That's it, but I'm out of ideas how to do it. Charities are not good enough; you need to disconnect the incentives for writing the stories from how readers like them. News, after all, is about observable reality so it shouldn't be affected by the moods of the society.
Yeah, BBC may be a step in the right direction. The grant-funding organization idea is interesting. Or maybe the reverse? Let's make news just like a public agency - collecting data (stories) and releasing it as a public service, just like NOAA does with weather? But then again, we'd need a way to incentivize the reporters to dig into government/whoever-funds-this problems and make sure that such reports get published as well.
It's a hard problem, and alternative solutions won't be perfect either, but the current situation of constant outrage-fueled brainwashing is taking a heavy toll on the ability of our society to reason, so IMO it's worth considering a change.
BTW. I think that a regular change of such systems may not be a bad idea. The problem with media, or with justice, or democracy, or even with markets start when they're getting too optimized, and thus start to deviate from the original purpose. So i.e. if you look at the arguments about why democracy is awesome and all, they all seem to make sense... if applied to the system from 100 years ago, back when it wasn't so thoroughly gamed. We want our systems efficient, but not too efficient, because they're not perfectly aligned with our shared values.