Wokeness is a mob, masquerading as "care and concern", but just a mob of often quite vicious and unpleasant people. People who are woke or happy to buy into the jingoism do it in no small part to use the dog-whistle accusations to set the mob onto people they don't like. I believe this is why Lloyd is "woke", and why it's pertinent to his behavior.
Being non-woke just means you aren't an obnoxious prick due to being woke. You can still be one for other reasons. A little like how not all democrats were KKK members, but all members of the KKK were democrats.