Congress shall make no law respecting an establishment of religion, or prohibiting the free exercise thereof; or abridging the freedom of speech, or of the press; or the right of the people peaceably to assemble, and to petition the Government for a redress of grievances.
That is not a definition of censorship. Nice try, though.
Regarding your first thought experiment, we already have that in the form of libel law. If you have standing to show that you are harmed by what is said and can show it to be intentionally and substantially wrong, you have a good case. Who brings the suit against general purveyors of fake news in your thought experiment? The United States Government?
The bigger problem is that when you move beyond the trivial, the dumb garbage that some people believe because it suits their biases and they don't care to find the truth, you quickly get into territory where people simply cannot agree on the facts. I believe that the truth can only be arrived at in an environment that allows free inquiry and open discussion. If you really think that the truth is a clean thing, and can't name at least a half dozen "open questions" in current world affairs, then you are not paying attention.