This is probably the most interesting look at the different sides in this debate that I’ve seen (link, not a quote, because the whole article is quotable). Essentially, Hickey argues that it’s a generational thing: adults who grew up in the 1970s when the repression of free speech by the government (in places like the UK) really was a thing. Versus kids who’ve grown up exposes to the endless toxic cesspit of “ideas” online, and just want a bit of space for once.
I think this argument is interesting in part because it represents a generation shift in the assumption of who has suppressive institutional power in a society. For Boomers and Gen Xers, this is very definitely “the government”; power, particularly illegitimate power, is always some kind of external, cohesive unit that can’t be escaped per se and must be fought. I think Millennials recognise this as a factor, too, but they’re also more likely to ascribe suppressive institutional power to individuals (or groups of individuals) within a society as well, which is where we get the whole “check your privilege” thing.
Of course, one of the big differences between institutional power and individual power is that the approaches to combating the illegitimate use of that power have to be different. You can overthrow or vote out or protest a government. You can’t really “overthrow” Reddit trolls. But you can escape them (or no-platform them) in a way you generally cant’ with something like the government.
I think most Millennials understand this distinction intuitively, and do so in a way that a lot of people who grew up in earlier political climes often struggle with. In other words, free speech and no-platforming are both tools, and they’re tools to be applied to different circumstances. But neither one in-and-of-itself is an inherent, ultimate good.