I've only ever seen it used when it's describing a woman who has resorted to going in all-guns blazing as a first recourse, and clearly has no idea what they are ranting about despite talking as if they are the world's leading authority, i.e. the sort of 'customer is always right' moron that you meet constantly if you have the misfortune to work in retail.
Personally, I don't have any respect for people that don't show any themselves, so if you come wading into a shop, shouting and bawling about your rights, repeating complete garbage as fact, and generally just being an utterly obnoxious human being, rather than making your point in a respectful and civil manner the way that other human beings actually deserve to be spoken to, then I couldn't care less if your behaviour gets called out for what is and somebody labels you a 'Karen'.
When it's used inappropriately then it ceases to retain it's original meaning and purpose. "Troll" has fallen victim to exactly the same thing. It sucks for people named Karen that this has gained so much traction, but it's undoubtedly a fact that the people that the term was originally intended to describe exist, so while it's lazy and offensive to people named 'Karen', I'm far more annoyed by the behaviour of the people the term describes and I'm not surprised that someone came up with a term for them and it has stuck.
FWIW I can think of countless examples where I encountered men behaving exactly the same way, and they're usually described as having 'little, fat, angry man syndrome', as they invariably fit a stereotype as well. So while there might not be a term for them that abuses a specific real name, they're recognised, spoken about, and held in exactly the same disdain as 'Karens', so I don't believe this is a thing driven by misogyny at all, at least, not originally.