Is that true? I've heard the claim before but not seen "proof." It's certainly possible however. For example, my own mobile provider routinely warns me if an incoming text or call is "suspected spam" or similar. But that is quite different from outright censoring a message.

[The following comments apply to my admittedly summary knowledge of US law.]

The law is "supposed to" make a clear distinction between "publisher" and "public forum" for computer services. We also have the much older concept of "common carrier" which applied to old-fashioned telephone systems, for example. A "Publisher" has the right to moderate content and (probably) more liability if he doesn't. A Common Carrier/Public Forum on the other hand, is supposed to NOT moderate content. So, in theory, the local phone company, or your cell provider, should not censor or cut off your service merely because you are discussing the next Klan meeting, or almost as bad, a voter registration drive for the local Republican candidate 🤡. Or to use another example, consider the difference in the service provider's ability to monitor the use of his service, and take steps to prevent illegal use of it: Two terrorists are discussing how to build a bomb and place it at a target. In the second, a publisher publishes a book telling exactly the same information. The latter example has actually happened at least once in U.S. law and there were court cases. The publisher was found civilly liable. In some cases, probably criminal convictions could arise too. But those are two examples of older, 20th century tech: a telephone and a printer/publisher. Now consider how communication has multiplied since then. Just who, indeed anyone, would have an implied or explicit duty to moderate content (e.g. censor) and face liability if they don't, in a case such as where an electronic document, an ebook, is publicly shared on a server? Or even a blog service such as Substack? Point: the duties, responsibilities and liabilities change quite dramatically, don't they? For example, consider the perspective that what your're reading right now is a living document (the article and its body of replies) subject to change at any moment. Sure, Substack has its use policy, but if I or anyone published certain speech that crossed a line, it would likely remain here for a certain period. If so, is it Substack's fault that they didn't have a censor verifying every post 24x7?

If I'm a spammer abusing an email service to send unwanted advertisements to hundreds of victims, is it a violation of my free speech if the service provider limits or cancels my account?

A major criticism of big social media has been they claim to be a public forum, yet clearly often muzzle certain speech, suggesting they are more of a publisher.

I claim no expertise in any of the above. My point is to show it's a complex issue, no doubt made even more complex by technology which often runs well ahead of changes in the law. I suspect that much modern communication such as social media cannot be distinctly classed as one form of service or another, since it probably has aspects of both, or even entirely new and currently unregulated ones.