She was asked a question about it once by Fox, because Trump had brought it up, and answered that yes, she would follow the law and not deny medically necessary gender affirming care, and noted that the Trump administration had also followed the law in the same way. It was a response to an unfounded fact-free attack, prompted by a question from an unfriendly network. She didn’t bring it up again.
You have not found an example of a leftist politician chasing barely-existing things in this example—the opposite, in fact, she was playing defense to right wingers trying to make something out of nothing at all.
Do prominent democrats do this, on some topics? Probably! But they don’t have a media machine and strategy structured around that as a core activity.
[edit] I mean, they do this because it works, of course. Look at the thread on PG’s piece, and this one. It’s clearly working to get people riled up and shift the zeitgeist, reality be damned. “Welfare queens”, that was a fun one, and so successful that I bet 35+% of Americans who’ve heard the term still think it was an actual problem. Some fizzle (“they’re eating the pets!”) but they’re not punished for those instances, so why not endlessly throw out BS and see what sticks? Some of it does, and then we’re all talking about a bunch of basically-fake grievances instead of anything that matters, and they may even use the BS to advance positions that do affect things that matter. It’s so very tedious to deal with.