Hmm, the thing is though, this info is rarely 100% false... people have tried things like labeling posts with truth contents, etc. and it falls flat. In cases where it is 100% false I think your approach works (although it might not actually be a sufficient deterrent).
In practice they always seems to be mixed with partial truths (when you read past the headline)... plus, determining what's "true" is kind of a nonstarter, especially at that scale.
I am also certain you would find things on NYT, WaPo, etc. that is not 100% truthful, even if the consequences are not as egregious. It's not just Breitbart. I say this because people on the other side of the aisle will take whatever you design and throw it back at you.
Identifying critical thinking as the problem (as you did in your post) is IMO somewhat right but maybe too hard of a problem to solve (as you pointed out).
The way I see it... we're in the middle of a tug-of-war between the old guard that used news & other power structures to broadcast what they wanted, and the new, democratized users who felt they didn't need to trust them while simultaneously being empowered to have their own voice heard.
The former would manipulate our perception of what was going on through news (print / media), while the latter are empowered by companies that control the new media landscape (Twitter / FB / IG).
What's interesting about the latter is that they leverage a system that was designed innocuously ("serve better ads") but can be adapted to control someone's perception of what is happening in the world (targeted content / engagement).
As of late, I've been thinking that the way forward is going to be identifying the forces that drive extreme polarization and re-imagining them to reign them in.
When I look at what happens with the way feeds are designed, there is such a crazy reinforcement of what I already "like" that drives polarization that maybe it really is just as simple as re-designing the feed to promote more diversity of content. It could be enough to temper some of the extreme outliers of craziness we see on either side of the aisle.
I will admit I am arm-chairing a bit here...