There are left-wing critics of "Woke", see for example the African-American Marxist Adolph L Reed Jr – https://newrepublic.com/article/160305/beyond-great-awokenin...
If an unapologetic Marxist is attacking "Woke", that really disproves the contention that it is purely some right-wing bogeyman
Or, consider that the Trotskyist International Committee of the Fourth International published a review of the sitcom Abbott Elementary, which includes the line "In fact, in its treatment of Jacob’s wokeness, Abbott Elementary refreshingly mocks the suffocating trend of racialism in American culture" – https://www.wsws.org/en/articles/2022/03/01/abbo-m01.html
Similarly, read their review of John McWhorter's Woke Racism – https://www.wsws.org/en/articles/2023/06/14/ihjm-j14.html – in which they largely express agreement with his criticisms of the progressive "woke" ideology, but simultaneously condemn him for making those criticisms from a pro-capitalist instead of anti-capitalist perspective
And see the socialist publication Jacobin's approving review of the philosopher Susan Neiman's book Left Is Not Woke, which attacks "wokeness" from an explicitly left-wing perspective: https://jacobin.com/2024/07/wokeness-left-ideology-neiman-re...