Getting back to the thesis, and without commenting on woke/not-woke (because I'm neither qualified nor care,) I am not sure what makes this particular comedy different from comedy we might otherwise consider problematic.
${X} joke about ${Y} -- if ${X} is only funny so long as ${Y} doesn't violate cultural taboos to make fun of, then we have an issue with the permanence of the joke. If a bad joke about ${Y} can be fixed by changing ${Y} to Republican (or whatever else is culturally acceptable to make fun of in time ${Z}, then the problem is not ${X} joke, it's our sensitivity to ${Y}.
${X} joke about ${Y} -- if ${X} is only funny so long as ${Y} doesn't violate cultural taboos to make fun of, then we have an issue with the permanence of the joke. If a bad joke about ${Y} can be fixed by changing ${Y} to Republican (or whatever else is culturally acceptable to make fun of in time ${Z}, then the problem is not ${X} joke, it's our sensitivity to ${Y}.