However “people who are offended by everything” is often used as a synonym for left leaning folks (or rather folks in favor of societal diversity; SJWs if you will). Also “shutting down the debate” is often used to complain about when a left leaning person reacts offensively (or even angrily) during a debate. This is regardless of if the preceding comment was actually very insulting or threatening to some people that may be present.
In a sibling comment I explain that—in my opinion—it is actually a good thing if people that hold oppressing and insulting views self-censor after having received angry replies to their offensive views. I‘d like to add now that normally us “left leaning folks” don’t simply label speech as offensive and leave it at that. We—as grandparent points out—we actively engage in the conversation and point out (sometimes in anger) the flaws in the opposing opinion, explain why a thing is offensive and bad, and why we are angry about it. Then we hope that either they will change their view or at least reconsider before they say something like this again.
Again I should reiterate that I am specifically talking about debates that I consider threatening or offensive to some groups of people.