I don’t think it was meant only as “watching out for the White man” so much as just simply being aware of the particular obstacles we encounter when navigating through society.
But, as with a Black guy taking a knee during the so-called “national anthem” prior to a sporting event, White people came along and turned it into something completely watered-down and not-at-all having anything to do with what it was originally intended for when Blacks did it.
I can’t speak for the White guys who rant and rage against “wokeness”, but my interpretation of that is they’re ranting against so-called political correctness, as usual.