I sense that I do not entirely grasp your line of thinking, but I will do the best that I can. Concerning the point in your original paragraph 2, I think that the old left was generally against censorship, but the new left has simply rejected those ideals, as opposed to not living up to them.
Why? What is it that makes you believe the 'new left' embraces censorship? This probably hinges on what 'censorship' means, but your cited passage argues that protesting speech is a good reason not to censor speech. Logically then students simply protesting the speech of others cannot in and of itself be the criteria for what the left is doing to censor education even if one did accept that college activists are representative of 'the new left' (as opposed to actual literal representatives).
What I most often see is people holding academics claiming academic protection but refusing academic rigour, then when their discredited ideas of little to no academic value are pushed without new evidence supporting it, they cry about being 'cancelled' or 'censored'. You would not advocate for not having standards at all, correct? What evidence should an institution accept is enough to allow a lecturer to advocate for forced-sterilization eugenics for example? Racial supremacy? These ideas aren't just unpopular, they're wrong as shown by evidence, and harmful as shown by history. Likewise I see people arguing that bringing in speakers of little credibility but highly contentious isn't worth the resources which can be spent elsewhere accused of 'censorship'. Now the perils of the 'heckler's veto' are known, but arguing the exact point of that line isn't 'censorship'.
Regarding your question at the beginning of paragraph 4, my current position is that if there were many more incidents of censorship coming from the right than the left wing, and if a right-leaning commenter said that therefore nothing should be done about the former until the latter were addressed, then I would be strongly inclined to assume bad faith. I would like to think about this some more.
Regarding the number of incidents of academic censorship, I heard Greg Lukianoff (president of FIRE) speak a few years ago, and IIRC he said that more came from the right than the left. However, the only recent examples I can think of at the college-level come from the left. High schools are a very different matter; Virginia's recent battle over Toni Morrison's novel Beloved comes to mind.
Well your chosen expert seems to know more than you would, right? Selection bias plays into what examples you can think of, and right wing educational institutions don't tend to get media coverage for their censorship as that's baked into their reasoning. And of course the kind of 'censorship' that can debateably be a label you could apply to some of the actions of left wing students and administrators isn't the same in magnitude, number, or kind to the legal efforts and successes from the right wing. Even just moving one's set to like for like, this 'public pressure' cancellation, the people screaming at school boards and making threats are overwhelmingly right wing. It's not like it's a close call here.
Regarding your second question, I would say that it is more disappointing coming from the left for the reason I provided above, not that it deserves more focus. I have participated in these two threads only sporadically, but political violence strikes me as being at the very far end of what I would classify as cancel culture, although I am not able to find a bright line.
If one, say ahhell, is bringing up threats of violence and violence as an 'escalation' of it from the left, then it would be rational to compare, right? And of course the actual risk from the threats should be.
The right are the ones actually killing people. Ahhell's argument is factually incorrect as johnny karate noticed. There are just not 'one of these for every one of these'. The right actually does shoot and bomb doctors. They pass laws requiring doctors lie. They shoot up Wal-Marts and federal buildings and synagogues and politicians and storm the Capitol at such a greater rate than the left that the threats from the right are of course, factually, much more credible. They threaten election officials and the police just pretend they can't do anything about it even when the reporters can find the people making the threats trivially easy.
So if the violence and threat of violence is related to cancellation as ahhell contends, it still isn't an 'escalation' by the left as the right has been doing it as long as I've been alive.
Or, as others have noted, 'cancelling' is just the vague term latched on to to define into existence a set of 'problematic' behavior that is deformed to whatever it needs to be in order to criticize left wing speech and exclude right wing. It only became a problem when the 'wrong' people got access to the power others have wielded forever.