Note: I am sharing a perspective to try to drive to the answer so bare with me. Secondly, realize I am not using toxic femininity as a Tu quoque fallacy
From the perspective of the male, masculinity embodies, societal expectations and traits that are unique to the gender. Why it is both (from the perspective of a man) is because men do not have a physical and biological markers as women do, which is a period and the ability to birth a child.
As such, masculinity is highly tied to society, the actions, and the traits of the gender as defined in society. Being a man is highly abstract, which maybe a foreign concept to a human being that is not a CIS-male.
That being said, women (aka tom boys) with high masculine spirits and trans-men can dance within this realm and be impacted. But as some people may declare men can not truly be feminists (not something I believe in) because they lack the ability to know exactly how women feel and be due to the unique biological markers and societal expectations.
If we take your pussy example, a male trait of independence and stoicism is being applied to mute with violence against himself to kill the emotional boy within the man. It combines the rage and emptiness that a man feels because he is not a woman and may feel "disconnected" since he at a base level does not see himself giving life.
So Toxic Masculinity will be highly personalized by men because of these linkages that I discussed. For women, it is foreign because they know with hard lines of when they cross the rubicon of girl into womanhood. So this is why the use of the phrase is highly dangerous unless a man is truly processing his masculinity and for some with therapy. I prefer the use of toxic culture of Masculinity, since it provides a a definitive barrier reducing the risk of personalization. IMO, my suggestion is a conversation started, the pop-culture phrase TM is a conversation ender and worse actually empowers the worst of the devils.
So the use of the phrase Karen and Toxic Femininity is just as bad in some ways as the use of Toxic Masculinity. It is what happens when the definition is good enough to get close, but the side effects from it allow it to be co-opted in the worst possible way. There is validity in all three phrases, and the most important thing is it is to allow differing view points the ability to work through conflict. It is meant to have a discussion starter.
My male view shows a Karen to be the purest form of Toxic Femininity, because the women weaponizes her female privilege in a patriarchal system to do harm to her gender. It is like Epstein's mistress who brutalized children. The wrong a man does to a woman is infinitely evil, being of the same gender makes it even worse.
Men feeling powerless in the discussion, then will grasp on to any powerful word weapon and use it brutally. So if a man uses the Karen mantra, he is embodying toxic masculinity in it's purest form. Women that do the same thing are also executing toxic masculinity. The thing is though, if one looks at this horrible situation through the eyes of a TERF (I am not implying you or I are these things). Men can not be women, and women can not be men, so ergo the toxic culture of femininity is the issue.
The real reality is the evil and vile actions are not gender specific. The corrupt and taint the unique and wonderful things of both genders. They are "toxic". They empower the negative traits found in the gender and echo/amplify through the societal makeup (currently patriarchal)
So I agree with you 100% that the weaponization of the phrase to shun and shut down women is wrong. That is not my intent either. The point is due to a lack of male communication and involvement in discussion on masculinity. The point of view of women who have taken a male psychologist point of view and co-opted it and reapplied on the gender actually will be interpreted as toxic to men. My proof to the claim above is read in society today about toxic masculinity and look at the ratio of authors female to male. You will see a pattern.
Toxic Femininity being talked to brings out the exact same feelings with women about gender and the societal issues and one can be concerned because of instead of fighting the evil, we are dancing around words. The core thing is, the raw distaste a woman has when the TF phrase is executed, is just the same one a Man feels. The thing is women have built systems to talk, and work things out. Men let it bounce behind stoic walls, and with only two emotions per patriarchal standards, anger and lust, get transformed. The other option is suicide or death by homicide (Las Vegas shooters and boys shooting up schools)
This is why Toxic Culture of Femininity is a better phrase, but it should not be a public mantra to talk to, since it will be weaponized against women. Just like Toxic Masculinity currently is weaponized against men, who are shunned, and feel shame and are muted like women were at the start of the Woman's suffrage movement. Karen is even worse because it doesn't even define what it is. It just makes assumptions which like beauty are in the eye of the beholder.
The big takeaway, and why I don't personalize it now is because I actually took the therapy and worked this for several years prior to the apex moment of #metoo and the Gillette commercial who used it to sell razor blades.
My healing to become the man I am truly began because I became a feminist. I spent decades processing true trauma and shattered masculinity and sexuality through a pure male POV to fail every time. Once I flipped the switch and looked at through a feminist eyes and read of the horrors of women and talked to them. I realized that it allowed me the perspective to handle my issues.
It is why in my poetry (which you may like) I talk to me as a CIS-Male, talking about my female and male warriors in my psyche protecting the sensitive glass boy.
A CIS-female has a glass girl with both those same warriors. The key is how to empower the male warrior (masculine spirit) to be equal and complimentary to the female.
The toxic aspect is societal expectations and taking the weakness in traits within the specific gender and identity (which includes race too) and distorting and breaking it down.
We need to fight that demon, not empower it. If we don't, the genders will grow more distant, sex and intimacy will be only seen through a glass wall with not touch and feel. Then women will join men in the equality of insanity that most men feel. The great disconnect from society and gender. It is a horrible experience, so we need to break those mirrors and glass barriers through words and action.
Words though have to be very very precise since they can do a lot of harm to those with little trust.
I am sorry for the length, but IMO these are the discussion more and more women and men should be having.