I don't think these two concepts are in opposition. Of course, exposure to a toxic ideological viewpoint is poisonous for the recipient, and may in fact entice them to become more toxic themselves (if they are swayed by said reasoning). But a crucial part of the societal contract is to tolerate such toxic viewpoints so as to have the opportunity to redeem those who hold them. In other words, by eating a little bit of the poison ourselves, we are granted the opportunity to save that person from their own poison, by showing compassion and empathy towards them.
People who have hate in their hearts are not entirely bad people. Everyone was once a child, everyone has experienced the mystery of simply existing, almost everyone has experienced love (whether from their parents or someone else), most people enjoy art and music. Evoking our shared humanity and showing compassion for others allows us to break down their toxic ideologies and even reform them to be beacons of compassion themselves.
Perhaps the best and most compelling anecdote to promulgate this way of thinking is Daryl Davis, who, through his talents and love of music, has reformed many of the most hateful people in our country — members of the KKK. I would recommend this TED talk by him, titled "Why I, as a black man, attend KKK rallies": https://www.youtube.com/watch?v=ORp3q1Oaezw. It really encapsulates the power of this viewpoint.
The crucial element here is ensuring that we have channels of communication that allow for this sort of compassionate understanding... I fear that the compression of thought foisted upon us by social media platforms and online communication tools obscures the basic reality of our shared suffering, joy, intrigue, and humanity. These important facets of communication can be lost to the aether of technology, and our true intentions are thereby belied by the incorrigibly weak substrate of written language.
As a note, I think we have discussed the topic of free speech in another thread before. It's good to "see" you again :)
Apocryphal means that the origin of the story is doubtful.
I think you might mean "the adage is incorrect" here.
On the other hand, similarly, an extremist exposed to non-extremists' propaganda potentially may be affected by it as well, decreasing the amount of overall extremism.
Or perhaps the total amount of extremism stays the same and instead it is spread out over large number of people, removing "spikes".
I do not know if it is true. But just an idea to consider - the discussed process of de-platforming removes _two-way_ communication method.