Generally, the older generations are all coralled in Facebook and twitter, the rest of the internet skews young. I tend to dismiss terms like woke out of hand. It's just online yoofspeak, not to be used by sensible adults and certainly not meant to be said out loud, unless ironically. However, much like its sjw and pc brigade ancestors, it's used by pricks to try to discredit and silence people who tell pricks to stop being pricks. The sentiment being "I have freedom of speech so you have to shut up!".
The whole transgender thing is just another item on the long list of things I don't really understand but just accept. Like quantum physics, or the Kardashians. It's not hurting anyone*, so why get worked up about it.
Live and let live.
^Basically this^
What find even harder to understand, is why the likes of Linehan and Rowling, whose audiences I would assume skew liberal/progressive, are willing to sacrifice their reputations and credibility to knock a minority down a peg?
There are of course some sticking points. World Rugby recently banned trans-women from competing in Woman's rugby. Can there be a balance between inclusivity and safety of competitors? Like I said, I don't understand it well enough to comment.
*Quantum physics is probably a bad example, it hurts my brain. Also there's a trend of new age hacks slapping the word "quantum" on their snake oil, but I'll find somewhere else to grind that axe.