Your assumption that this is only a problem on the left is completely baseless. The right likes to use the freedom of speech argument when they feel it works against them, but they also support scare tactics to suppress the speech (and votes) of minority groups often with the support or tacit acceptance of the government.

It doesn't appear that the right even understands freedom of speech. Freedom of speech does not mean that private organizations cannot censor you, it means that the government cannot censor you. Further, freedom of speech does not mean freedom to espouse hateful views, advocate physical violence (as you said), or cause false panic (whether you yell fire in an airport or yell conspiracy theories on the radio). If you believe that we should have open platforms on the Internet free from the censorship of Internet giants, then that is a different issue than freedom of speech.

It's not that the left dislikes freedom of speech, it's that the right has twisted the meaning and use of freedom of speech to promote hate, conspiracy, and fear. I recommend you check out the Paradox of Tolerance, and idea put forth by philosopher Karl Popper.