Then why is this specifically encoded in the first amendment?

The amendment is "Congress shall make no law...". It is solely a prohibition on government power and nothing else. You have guaranteed free speech when a citizen interacts with the government. You don't have it when interacting with a private institution. This is how social media can censor and deplatform with impunity.

From the text: "The exceptions here or elsewhere in the Constitution, made in favor of particular rights, shall not be so construed as to diminish the just importance of other rights retained by the people, or as to enlarge the powers delegated by the Constitution; but either as actual limitations of such powers, or as inserted merely for greater caution."

From the text: "The exceptions here or elsewhere in the Constitution, made in favor of particular rights, shall not be so construed as to diminish the just importance of other rights retained by the people, or as to enlarge the powers delegated by the Constitution; but either as actual limitations of such powers, or as inserted merely for greater caution."

This is the correct response.