Tail end of the '00s and early '10s was about the time we had a new wave of "words you can't say" (how many words do we obliquely refer to with "x-word" constructions now? At least three or four fairly-common ones, right? There was really just the one, before that) and other changes like male-as-neuter pronouns going from taught as correct in school (through all of the 90s, at least, and probably a bit beyond) to "may get you scolded" (2010) to "will probably get you scolded" (2015).
We've been sensitized to so many more words than in, say, the year 2000, that even for one who doesn't have a dog in this fight, watching early '00s media (let alone earlier) can be distracting because it's hard not to notice the now-red-flag words or topics that are thrown around casually—especially in standup, or on comedy TV shows. It's been a pretty fast shift, and I'd put the start at the tail end of the '00s or early '10s.
I mean, hell, just look at where mainstream Democrats stood on gay rights in the early '00s, for that matter. We've gone from, "gay marriage? I dunno, that might be a tad too far" to "we must protect trans rights" in a remarkably short span. I dunno, maybe the OG civil rights era had a similar feel of The Discourse shifting a ton in only a decade or two—I wasn't there for it.
It's probably also worth noting that this shift looks rather more total from a very-online perspective than it does IRL, which is probably a source of some of the ongoing friction. Online, it's done, the shift has happened, you're a cave-man if you're lagging behind on it at all—offline, much less so.
