Most of the political violence in recent years has come from the left, but I think that’s more due to ideological capture than anything inherent about left vs right.
Because the left captured all the institutions — tech, media, academia — people on the left were able to live in a bubble. They were able to avoid the unpleasant cognitive dissonance that comes with encountering opposing viewpoints. It was like being in a very large cult.
When this persists for long enough, it’s easy to become a fundamentalist of sorts, someone who views contrary viewpoints not just as incorrect but evil. It’s easy to see how this would happen — as your worldview gets ever more affirmed, it becomes ever-more painful to have it exposed to contradictions and internal incoherence.
People on the right the last few decades were constantly told they were bad and wrong by the media. It was almost impossible for them to live in a bubble. When you don’t live in a bubble, you are exposed to cognitive dissonance all the time, and you have less absolutism and fundamentalism. Less existential dread when someone disagrees with you, more rigor in formulating your worldview.
What the right should not want is to swap places with the left where their views become fundamentalism, dissent from them is verboten and their side gets radicalized when challenged.
Free speech you disagree with has to be protected no matter what. Unfortunately, I think it’s just the nature of power that it corrupts, and when the right re-takes it (they have in the US, but not entirely as the institutions are still very much leftist), they’ll probably do stupid shit like outlaw mocking Charlie Kirk’s assassination as “hate speech” or outlaw flag burning.