I think calling someone a Karen is a good way to try and shut down someone’s point of view while signalling that by default you are certainly no Karen yourself, therefore you must be correct, i.e. let’s all have a good giggle at the other person while we pat ourselves on the back for being morally superior, more socially aware, more insightful, more caring, social justice warrior etc.
I find the nu-left has certainly embraced the Karen principle with gusto, possibly around the same time it embraced cancel culture, virtue signalling, dismissing free speech and stopped trying to listen to all points of view in order to find common ground to move forward. Is it a quick easy way to dismissing another point of view, trying to discredit them without needing to argue a point?
Just by saying someone is a “Karen” might make others hesitant to agree, just because they don’t want to be seen in that light themselves. Possibly it’s also a good way to create othering, so a good guys v bad guys situation, where you never actually need to argue a point as you’ve already staked your claim subconsciously.