We have clear examples of authoritarian states successfully suppressing specific topics. It's not so much about killing it as creating a line where consequences start happening.
China is one example, where being too vocal about certain topics online can land you in a police station interview.
I don't think we should be overly confident that similar things can't happen in the US or EU.
And indeed, in the US we're seeing the administration successfully pressure changes at educational institutions for example. We're already uncomfortably far down this road imo.
The right is great at “killing” ideas or atleast the momentum for it. The left has been keeping true to the first amendment rights. It needs to start doing what the right does when it doesn’t like ideas. Somewhat violently shut it down. The right doesn’t care about free speech unless it’s for their specific goals.
Funny. In my five+ decades on this planet, I seem to find the right more apt and eager to employ 'thuggish authoritarian attempts' -- akin to my time growing up in East Germany. But now being in America for over a decade it's really not much different.
Usually these things end up bringing more attention to a specific group or cause due to the Streisand Effect.