In case of free speech as an ideal, it's still a bullshit argument. You cannot suppress speech with speech alone. Go on any anonymous internet forum and try to suppress someone's speech by e.g. threatening to doxx/harm them - you will be laughed at, because on the internet there is no real threat of harm. It's always the threat of harm that actually suppresses speech, not speech itself.
That fact that you use speech to deliver the threat doesn't in itself create a paradox.
In context of freedom of movement, that argument would be akin to "free movement is a paradox because you can suppress someone's movement by holding them down". Yes, you use free movement to walk up to a person, but it's not your movement that holds them down.