I disagree. Freedom of speech is a human right, and rights exist as between people and the government. There are only a couple of instances (things like defamation, harrassment, and alike) where a citizen have a recourse against another citizen in a way that limits speech.
You'll get a couple of philosophers that talk about the need for speech to be unrestricted in public discourse (e.g., John Stewart Locke) but I can't think of any point in society when that has actually been in effect - usually, because the right deemed certain speech to be unconscionable ("promotion" of homosexuality, blasphemy, or anything else antithetical to "conservative values").
There was a stretch of about 10 years where "unconscionable" speech did not align with conservative values. At that point, the right (by which I really mean politicians and the media) suddenly declared themselves ardent defenders of free speech (previously more a liberal value). However, even during that period, many of the same folks who presented themselves as free speech champions either engaged in cancel culture or attempted far more radical ways to silence their political opposition.
This being a US thread, the right (and yes, the same outlets of politicians that called themselves free speech defenders:
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led high profile "cancel culture" campaigns - Colin Kaepernick was probably the highest profile cancelation of the 2010s, because he dared kneel during the national anthem to protest racism;
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removed masses of books from libraries;
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published lists of political enemies (e.g., Charlie Kirk's organization published lists of university professors who espoused left wing views, who would then be subject to harrassment and death threats;
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extremists engaged in widespread political violence and murders - if it does prove that Kirk's killer was informed by left wing views (as seems likely), then its one of a tiny minority. 95%+ of political violence and killing being committed by the right (although now, in the wake of Kirk's killing, its suddenly objectionable).
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the big one - they tried to overthrow democracy, to extinguish the most important act of expression in a democracy- the right to vote. Free speech defender Charlie Kirk was part of that.
And, of course, now that they're (back) in power in the US, they're pursuing the most sustained attack on free speech seen in our lifetimes, with many (not all) cheering it on.
So yes, I find it impossible to accept that large segments of the right - particularly those who engaged in cancel culture or supported widespread and sometimes radical attacks against freedom of speech - genuinely and sincerely cared about free speech, even if they loudly and regularly purported to.
Obviously, that doesnt apply to everyone on the right, but it does apply to right wing politicians and media in the US in general.