Sometimes there’s a bit more nuance, but it does tend to come down to this. Often flagging mechanics are pointed to as censorship, or anything which falls short of engaging with someone in debate is censorship. In most cases there’s a lack of understanding that just as they have a right to say something, other people have a right to respond with a flag, or not respond at all. Freedom of expression doesn’t imply an obligation to debate everyone on every point. I’m not censoring an anti-vaxxer by flagging and moving on, I don’t owe them a debate. The government isn’t allowed to interfere in the right of anti-vaxxers to spew, and also in my right to call them morons, or a social media site’s right shut the whole conversation down.
Too often “free speech” is a messy and conflated concept used to basically argue against anyone being told to shut up. Various arguments about the size or dominance of a social media platform are often used to try and get them covered like the government under the 1st amendment. Then if all that fails, the argument switches to free speech as a vague principle rather than a legal right.
To be fair, it isn’t just conservatives who seem to feel this way either. Americans in general just seem, on average, not to understand this issue as well as they think they do.