Yes, I'd say that's the size of it.
Oversll, they are people with a limited capacity (or education) to reflect on their own privilege - like fish who don't understand what water is.
They are people who think that the high level of comfort they feel in society, just feeling OK saying whatever is in their heads, is normal. They don't notice that it is a privilege. So they feel that suddenly being asked to guard their tongues, or shift their way of being, is a new thing - an insult to their freedom and ability to operate in the world. And it does feel like that, so I sympathise with those people's sensations.
However they don't understand that other people (who are different from them) already were living in an unwelcoming world where they couldn't speak freely/have been judged/are constantly harmed by culture and its assumptions.
What "woke" does is redresses the balance.
So, some people feel a but more awks in the world for a while...in order that others, who have been, historically, systemically under-valued and felt less comfortable, can feel... a bit more welcome. A bit more normal. A bit more ok for being themselves.
This is misunderstood though and when you say "look at your own privilege" anti-woke people indignantly think "But I don't have privilege! I'm not rich! I work hard! I don't get something for nothing!". This isn't what privilege is. Privilege is the ability to feel normal in the world without having to constantly think about what you're doing.
So the anti woke person says "People are such snowflakes they can't handle a joke about identity!" and marvel at the weakness and over sensitivity of others. They are not considering - literally not "awake to" all the work already being done by the person with the "less normal identity" just to get through the day. So the joke is the last straw, not a tiny nothing.