How would you propose to fix that? Should candidates only be allowed to make "true" statements?
Civilians should be able to trust the elected leaders' words.
In this case, Facebook is "the press". Keep in mind that there is no legal definition of what it means to be "the press" — rightfully so. NYT v Sullivan is particularly relevant because, in the case, NYT carried out a full page ad by supporters of MLK Jr containing several factual inaccuracies, such as the number of times King had been arrested during the protests, what song the protesters had sung, and whether or not students had been expelled for participating. The Supreme Court unanimously ruled that, unless you can prove "actual malice", the free dissemination of information is protected by the Constitution.
Those that post on FB are already held liable. The whole point is that Facebook Inc isn't liable. If you were to post the text of the first chapter of Harry Potter in a FB post, Facebook Inc would not be liable for copyright violation, it would be the individual (or association) that posted it.
> In my humble opinion, it should be perfectly possible for citizens to sue their politicians for being lied to.
It already is possible to do this, it's just practically infeasible. If you are able to prove "actual malice" — I.e. if you can prove that a politician knowingly lied to their constituents, they can already be held liable for damages. The burden to prove this is high enough that you would need to find documented evidence that the individual in question knew the truth and specifically lied. Remember, it is not illegal to make political promises that you can't keep, and it is not illegal to be publicly wrong about the facts.
> Civilians should be able to trust the elected leaders' words.
No disagreements there. The only issue is that the First Amendment makes it essentially impossible regulate how those words are disseminated.
That is to say: there may not be a way to objectively say whether a given statement is true, but it is possible to objectively say that some statements are false.
Science is the study of falsifiable theory. Things that cannot be falsified lie outside of science.
There's a big difference between "candidate X says that pigs can fly and this is why that's wrong", and "The flying pigs above your house are bigly bad", as far as public messaging goes. Social media has enabled the latter, and it's kind of an unsolved problem. Social media has also caused a major blurring between advertisement and content, which doesn't help.
EDIT: An interesting/worrying byproduct of this seems to be that parts of the media are going further and further off-piste, and letting complete _nonsense_ go by without the "and here's why that's wrong" bit. A glaring recent example was parts of the US right-wing media absolutely tying itself in knots to explain why Trump's rambling about disinfectants for treatment of COVID in humans made sense. It's just really hard to imagine this happening 20 years ago.
https://www.wired.com/story/journalism-isnt-dying-its-return...
The history of the printing press was fraught with exactly the kinds of concerns people appear to have with social media. In those days, you had politicians routinely troll political opponents, publish partisan screeds under pseudonyms, and openly advertise their own biases.
They were never regulated by law because the Constitution makes such regulation essentially impossible.
To be clear, I don't really see a problem with media being partisan as such. It always has been to some extent. The direct line from politicians to public is new, though (at least on this scale; there are prior examples of politicians using radio in this way, but it was somewhat more restrained).