I think when you look at the reason it's become popular as a pejorative, it's not that it wasn't a good term in the first place. If you look how it was used originally in the civil rights movement, it really describes a kind of psychological experience many people have had in one area or another, where suddenly some pattern is seen and it's like you see things through new eyes.
But the identity politics progressive orthodoxy people, who I think began to use it in that same way, are just so ridiculous in terms of their inability to see anything clearly, that it was almost inevitable they would be mocked using that term. Which maybe should be a warning, because they honestly think they are seeing some sort of great truth here that now informs all of their thinking, but then they come out with deeply racist crap like "be less white" or want people to undergo pseudo-scientific unconscious bias training.
But I'd really place the basis for what gets called wokeness in Critical Theory. The gender stuff really grows out of that in various ways, as well as a lot of their analysis of things like race. Unfortunatly a lot of feminist commentary has been based in CT as well and it's I suspect teh main reason all the women's studies departments turned into gender studies departments.
CT can really only make arguments or analyse reality in terms of power structures. Every time the main thrust of an argument is "x is more oppressed than y" or "No, actually, y is more oppressed than x" that is CT rearing it's ugly head. And I would argue intersectionalism is just the way CT tries to deal with the fact that people often fall into more than one of it's hierarchies. But the underlying premise about how power works is just wrong - that's why it's analysis fails. Intersectionalism doesn't really solve the problem.
As mentioned, CT is also very presecriptive about what people are supposed to believe and doesn't care much about free speech or even free thought. It sees freedom of human beings as its goal, but it doesn't mean freedom to think or believe. It means freedom from oppressive ideas and social structures. Securing that freedom can easily mean preventing people from saying or thinking bad thoughts.