The First Amendment is pretty much irrelevant on the internet. It prohibits only government censorship of free speech and a free press. It leaves others – notably social media providers – free to censor internet content, which they do routinely. Why do we tolerate this?
The first reason is that without censorship, the internet would be a sewer. Its creators may have originally envisioned it as a pure mountain stream that would allow knowledge to flow freely to all humankind. Unfortunately, the humans – well, at least some of the humans – came along and polluted it with content that many find objectionable. The internet became a cesspool of sexual explicitness, unmitigated violence, hate speech, etc. Censorship began as a way to repress the content most people didn't want to see or the law prohibited. Now, it's a habitual practice to which we are accustomed.
Of course, in the United States, Section 230 of the 1996 Communications Decency Act gives social media providers a free pass to host (broadcast-publish) whatever content their customers want to originate. Specifically, it says: "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." Unlike publishers, social media providers have no liability for the content – even objectionable content - posted by their subscribers.
So what's a publisher? A publisher is someone who produces published work, like a newspaper. They pay individuals to create content for their publications. Every newspaper has a different look and feel, cultivating an image - often using a motto - to convey its values. For example, the New York Times motto has been "All the News That's Fit to Print" since 1897. Such publishers have three goals: to print truthful stories, beat competitors to the reporting punch, and make money.
Newspaper publishers assume liability for the material they publish. They can become involved in costly legal proceedings, but the positive publicity generated for righteously defending free speech can only boost their reputations and bottom lines. Sometimes papers get into legal strife because they tread on national security issues while exercising their First Amendment rights. The New York Times and The Washington Post both faced some very sticky legal issues when they respectively chose to publish stories about the Pentagon Papers and the Watergate burglary, but their publishing prestige skyrocketed.
How is a social media provider different? Thanks to Section 230, social media providers have no liability for the content that appears on their platforms. They also don't pay for their content. They may do some cursory fact-checking, but reporting the truth is not among their mantras. Social media companies market themselves with platitudes of altruistic intention, using mottos like "to bring the world closer together." In truth, social media is really about making money.
This is the second reason for the wide acceptance of censorship on social media. These platforms make money through advertising. Advertisers will only (continue to) buy advertising if the social media platform demonstrates high ad viewing and click rates. If a subscriber starts seeing too much irrelevant or "objectionable" content, she may be motivated to stop using the platform. Since this threatens revenue, social media companies moderate - some might say censor - content to attract and retain customers and advertisers.
It is a myth that social media provides a 21st-century "speakers' corner" where a person can share her uncensored views. Social media platforms may pretend to espouse democratic ideals, but in reality, profits drive them like any other money-making institution.