Don't rely on cartoons (even xkcd) for accurate legal analysis.
If broadcasters voluntarily try to fawn upon the government, that's a different story.
Freedom of speech is a broad philosophy "that supports the freedom of an individual or a community to articulate their opinions and ideas without fear of retaliation, censorship, or legal sanction" [1]. It's a cultural norm and philosophical principle more than a legal one.
The First Amendment is a legal principle that restricts the government from policing speech.
So yes, if "you're yelled at, boycotted," banned of cancelled because of what you say, that is not consistent with the principles of freedom of speech. Whether your right to free speech has been abrogated is more complicated and context dependent, since rights are a social contract. And in no case is a private actor retaliating, censoring or sanctioning someone for what they say a First Amendment violation.
We tend to conflate the First Amendment with free speech because the former probably remains one of the strongest expressions of the latter in any body of law in the world. But it's important to remember that free speech is broader than the First Amendment, and that someone can have their freedom of speech infringed upon without falling afould of the law. (The fact that the First Amendment restricts the Congress from "abridging the freedom of speech" directly cites the fact that freedom of speech exists outside the First Amendment and that it's possible for someone other than the government to abridge someone's freedom of speech.)
[1] https://en.wikipedia.org/wiki/Freedom_of_speech
[2] https://constitution.congress.gov/constitution/amendment-1/
