Woke is a scare tactic from the right wing media. To scare people out of teaching their kids education.
I don't think that's necessarily a fair descriptor.
Not that I necessarily disagree with your sentiment that it's often used as a scare tactic, but where I'd disagree is on the aspects of what's being taught under the guise of "education".
Presenting it as purely "education" is the same semantics game we see on a lot of issues where one side will appeal to a sensible-sounding vague abstract (that most reasonable people would agree on), and then claiming anyone who opposes any part of their agenda must be in opposition to that abstract concept.
I can provide an example from both sides.
From the Right, their efforts aimed at promoting "Religious Freedom". When you look at the finer points of what they actually want, it's pretty clear that what many actually want is to put traditional conservative Christianity on a pedestal and force everyone else to color in those lines. (or at a more basic level, give themselves a legal way to circumvent anti-discrimination laws).
Would critics of conservative "Religious freedom" bills be "against religious freedom"?
From the left, labelling things like gender affirming approaches for minors and elective abortions as "health care", as a way to be able to claim "conservatives are anti-healthcare" when they predictably oppose it.
So while the original definition of "woke" may have been "being aware and sensitive to cultural and social injustices", in practice, a lot of it has become "weaponized political correctness", and that's what people people are objecting to. At it's core, very few people (even on the conservative side) would oppose teaching a more unvarnished account of history, and wouldn't necessarily object to other people wanting to be hyper-sensitive to things with regards to how they conduct their own behaviors. When it becomes a problem is when activist types say "Well, I'm going to teach it to your kids so that we can make sure when they grow up their on our side and not yours", or when the stance is taken of "you MUST go along with all of these things we want and don't dare go against it, else we'll go digging through your facebook posts from 4 years ago and find a reason to get you fired"
This was something that even Obama pointed out when speaking to young people (and in another speech, referred to it as something that ends in a figurative 'circular firing squad')