AH yes, it's all about politics am I right?
Fact: the left appropriated "woke" from the black community and changed the meaning and usage of the concept of being woke.. Historical usage of woke, wokeness, stay awake, etc whereas if you are dark skinned you need to stay alert to racism in certain places, by certain people, and in certain situations. White people who think that they are woke are in the same boat as other white people who engage in verbally attacking people of color. One can stay aware, especially of social problems such as racism and inequality even if you are not a person of color. But white people claiming to be woke are on the wrong boat.
White people cannot be woke.
It is a bit more complicated than political rhetoric between the left and right which ignores that racism is not a divisive left-right thing where one side is presumed racist and the other is not. Humans are humans, some learned to be racists and/or bigots and some learned not to be those things. The left version of woke though is just an attack on the right and does more harm than good for racial relations across the board. Your own response above proves that point. You framed racism as a division between left and right.
The theory of left and right is a global belief system, that is too ambiguous to be taken seriously. The left blames Nazism on the right and the right blames Communism on the left. Both of those systems of so-called governance share many aspects. And they go further; the leftist argument is that Nazism and Soviet, Maoist Communism are actually right-wing examples. While the extreme right asserts that not only are those types of Communism are leftist but so is Nazism.
They cannot both be right so logically they are both wrong.
And back to my point: It is in fact an insult to accuse someone of not being aware of racial injustices. Calling someone unwoke is the same as calling someone a racist. Calling someone a racist is a pejorative even if that person is a racist.
Calling racists, racists will never make those idiots not racist. Racists are proud of their racism so telling them that they are not woke is like a badge to them. And if the intent is to think that there are non-racists in 2022 that are not aware of racial injustice is incredibly arrogant lording at best. And add to that the need for non-racists to be on the left, outs the actual intent of division.
So yes saying that someone is not woke is pejorative/political rhetoric that does more harm than good to racial relations and the hope to live in a better world.