Democrats stopped using that word likely because seldom anyone actually used it in any significant way before it got co-opted by rhetoric that uses it to denigrate their critics. I seen it on twitter in the past but never on mainstream political discourse before it came out of Trumps mouth (in a way that crossed the Atlantic at least). Just being critical of issues that are not even controversial in Europe is enough to be tagged as infected by the woke mind virus on X/Twitter.
Coming from an "outside" view of American politics, the American left is incredibly conservative, both economically and morally so these labels are at best distractions and used by people to reinforce the duality of something that should be plural. This is a guilt that both democrats and republicans are carrying. Is just that Republicans are particularly nonchalant about what I view as anti democratic discourse.
I've seen something very similar in Brazil with the conservative parties in there adopting the "Petralha" (as in someone sympathetic to the Workers Party) or Communist as a way to denigrate any non conservative view independent of actual affiliation or opinion. Its not meant to be accurate, just mean to group all criticism together, because if you can do that, then its easier to use the stupid criticism (from radicals as you pointed out) as a reason to classify any criticism coming from "that kind of people" as stupid.
From the article shared (no the economist one) its clear that the definition adopted of woke by the author is cherry picked to be inflammatory. Likely real from some people but certainly not something that is shared among mainstream Democrats and I would bet not even a sizeable minority. Parts of it do resonate with the majority but thats part of the allure of using it, you create something obviously controversial that no one believes in full to tie everyone in the same bucket.