Most here at HN think of the left as wanting more censorship in order to avoid hurtful speech, but that's not generally how the left thinks of it. On the left, the view is that the practice of maximizing freedom of speech by tolerating extreme speech, has the effect of reducing the freedom of expression of the non-extreme, non-powerful constituencies.
In other words, protecting freedom of expression by politely tolerating hate speech has the effect of making the targets of hate speech less free to express themselves in practice. Like a heckler's veto, the presence of hate speech dissuades minorities and the less powerful from engaging at all. It's not freedom from being offended that the left wants, it's a less corrosive public sphere that doesn't demand that the weak struggle through a painful headwind in order to be heard at all.
Having a extremism-based standard for measuring freedom of expression creates a requirement that participants tolerate the material harm of extremist expression. The left wants to shift the standard of measurement to tolerating more speakers of different kinds, rather than more subjects of varying extremism. The left still believes in maximizing freedom of expression, they just recognize that there are different metrics that may apply.