Wow. Great find. Whoever decided to delete her post should be arrested, charged and tried for breaking the law.
The suit was a libel suit by the politician who was the target of the advertisement in the New York Times. The court found in favor the advertisers and the New York Times, saying that minor inaccuracies are not enough to discredit and punish the people from criticizing their politicians.
The parallel to the current Biden situation is that the advertising platform (Facebook) is taking the opposite approach from what the New York Times did in 1964. Facebook banned the mother from criticizing Biden, while the New York Times ran the ad criticizing the Alabama politician.
Regarding the notion that it's a private business and can do what it wants (not obligated by the 1st amendment), it's true. The New York Times could have refused to print the advertisement, but it did not. Facebook did refuse to let the mother post future criticism of Biden.
Also, I'm not sure how campaign advertising plays a role in this. In 1964, the New York Times might have been required to run an ad that was political during an election cycle. Print and broadcast media were obligated via regulation to air all sides of paid debate during elections. However, internet media are not included in these campaign regulations (they didn't exist when the laws were passed).
Call it an election advertising loophole that lets Facebook and Twitter censor opposing viewpoints that they disagree with, but broadcast TV and print media must accept paid political advertisements.
This is why broadcasters, for example, cannot refuse to air ads by Donald Trump, but Facebook and Twitter can.
-PJ