There is truth in the observation that, at this moment in history, conservatives use the rhetoric of free speech more than so-called "leftists." This shouldn't be surprising when male leftists are kicking women in the face for being pro-life. The Left has so much political power at this point that it's not enough for them to silence speech that they don't like, they're actually compelling speech.
History has shown us that conservatives, when they've had the power, have pushed for their own brands of censorship. George Carlin's "Seven Dirty Words" bit was an attack against conservative censorship. History is full of examples of conservative censorship.
What's wrong and disingenuous and counterproductive is to start to label free speech as a conservative value simply because conservatives are the people disproportionately, at the moment, using the free speech rhetoric. Free speech has always been, not only a liberal value, but the liberal value.
Believe it or not, there are those of us who see free speech as more than an expedient. There are those of us who bothered to educate ourselves on this issue and defend it whenever and wherever free speech is threatened regardless of who is threatening it.
There are those of us who believe that the first place to defend freedom is the first place it is denied. If you want to reject that value and label it as "conservative" I honestly don't understand your angle.