The moment Facebook chooses what to display, they own the content.
And it just so happens that what Facebook chooses to display, chooses to recommend, are the lies, more often than not.
As an aside, chronological sort key has been available in the previous designs, and the URL still works
www.facebook.com/?sk=h_chr
It has its limits of usefulness, though - some group admins moderate new posts in bulk once a day (if that), so depending on the time of visit you might be swamped with dozens of posts from the same group.
You might be getting confused by the concept of a common carrier. Facebook is not a common carrier. It might be possible to amend the common carrier law to cover Facebook, but in the current political climate that seems unlikely.
47USC230(c)(1) "(1)Treatment of publisher or speaker No provider or user of an interactive computer service shall be treated as the publisher or speaker of any information provided by another information content provider."
And definitions-- 47 USC 230(f)(3) The term “information content provider” means any person or entity that is responsible, in whole or in part, for the creation or development of information provided through the Internet or any other interactive computer service.
Of course, CDA also excuses Facebook from making tilted or faulty moderation decisions in good faith.
Why do we assume our own governments will also be immune from the same temptations?
As just one example, take Hunter Biden's laptop before the election. This got memory-holed so fast, half of America doesn't know it exists, and 2/3rds of the rest believe the 50+ intelligence agents who immediately said, with no proof, it's Russian disinformation. This was epitomized by the reporter who said, -even if the laptop is true-, the press must treat it as disinformation.
One doesn't know what one doesn't know.
The press cover people getting accused of crimes all the time. The press does not say "this person committed that crime." The press reports on arraignments, events at trial, verdicts, and reactions from relevant parties like legal experts.
The scientific world in general is a community where research about things comes in, people digest it, and come to a consensus. This includes some contradictory things or things that turn out to be faulty conclusions. This does not make them "fact check verified false". It is a stream of information that is weighed and honed over time with further research or questioning.
Now I agree people are very bad (even "smart" people) at ditching their inherent tribal nature to weigh data that supports their bias more highly. The echo chambers of partisans on social media will assign an impossible burden of proof against evidence that challenges their thought on something and have no burden of proof on the converse. But that does not mean that we can then make it binary of "true" or "untrue" to stop this. "Fact-checks" feel good, but it is clear now they are being extended to not just the realm of demonstrably false information and more into the opinion and inconclusive realm.