Interesting discussion.
It undoubtedly is an important word to black civil rights activism, but neither the word "woke" nor the concept behind it originate with the black activists using it today. It comes from the writings of post-modernists and post-structuralist, and found expression in various critical theories, around the 1950s and 1960s. Hence it does not merely describe someone awoken to the realities of racism, but it describes anyone who develops a critical consciousness around any aspect of social justice. (The consciousness raising sessions held by second wave feminists are another example of this.)
And the underlying concept, arising as it does out of critical theories and post-modernism is not without problem. (But nothing ever is when it comes to ideologies. Especially those intent on breaking and remaking the world in their image.)
Likewise "Karen" is not originally a word used to refer to white women in a context of racial prejudice but a misogynist slur coined by a very bitter man who posted about his wife, Karen, and how much he hated her. It took off from there. Later, black activists started using the term in the manner described. It always was and, I would argue even in the context of the discourse about race it continues to be a misogynist slur.
I think we should be able to at least consider that as the word woke is claimed by a number of other social justice movements, that there's some justification in Burchill using it the way she does.