I'd say quite a few, if not most, moderate progressives are not at all on board with extreme wokism. Similarly, moderate conservatives may not be too crazy about Trump or "alternate realities".
This massive group, which is most people, and pretty much all sane and reasonable people, have no place in social media like Twitter. The silent majority. They can't even express a single critical thought about their own "side" nor engage with political opponents. They're scared.
Rather than doubling down on this battle of which side should be censored, the very point should be to reduce extremism and its reach on both sides. Not just "your" side. Sane people should dominate conversation.
Yes, there are people who use the term too eagerly, just like there's people that use the term "alt-right" or "far-right" too eagerly. But we all know what woke means.
I'm from the deep south, perhaps you are too, but there is definitely a thing known as hate speech, and there's definitely a thing where the leaders of a town are happy to keep black people down, including using derogatory words. Their speech reflects their views on life. I disagree that we are improved as a society for them being able to make such comments in the public sphere.