No, it led to people calling for the resistance to the WWI draft being imprisoned: https://en.wikipedia.org/wiki/Schenck_v._United_States, https://en.wikipedia.org/wiki/Shouting_fire_in_a_crowded_the...
I think that if you draft your country's constitution to say, "We have freedom of speech, except for calling for genocide and/or denying the Holocaust," then that can work reasonably well, and it won't necessarily turn into a broader ban. If you actually write a short list of exceptions into your constitution, then that can work.
But the US constitution isn't written that way. We have very broad freedom of speech, and the only way to curtail it is via the Supreme Court deciding that certain kinds of speech are somehow an exception to rules. Libel and slander are, of course, exceptions, but they're ancient ones with a lot of case law explaining exactly what's forbidden. There are other exceptions, mostly involving things like angry mobs.
But if you want to start saying, "This speech is forbidden because it promotes bad ideas," then you need to fundamentally change how the Supreme Court interprets the first amendment. And once we go there, then we have no idea where we're going to wind up.
No, because we never banned shouting “Fire!” in a crowded theater; the idea that such a thing was unprotected speech which could be banned is non-authoritative dicta from a Supreme Court ruling which did, however, uphold he criminalization of core political speech in the form of protests against the WWI draft, which has since been overturned and is widely recognized as a major failure when it comes to free speech. Which was worse than a content-neutral ban on all speech in theaters.
Free speech is about being able to state your oppinion. The person shouldn't "Fire!" in your example doesn't have the oppinion that there is a fire.
Did banning shouting "Fire!" in a movie theatre lead to all speech being banned in all movie theatres?