With that male card we see companies where every male employee is automatically CEO and play golf and sit at the beach and all the lower level employees are female and forced to do all the work, like a scene from a slave plantation.
I can honestly say I have never seen this mythical company, and if it existed then the wast majority of the bottom 10% of the population would no longer be male.
You're building a strawman. An example of the male card is having every resume with a female-sounding name get extra scrutiny to make sure they're qualified. This is backed up in artistic fields by what happens when they move to blind auditions[1]:
> Blind performance auditions, much research has proven, often results in the hiring of more women and minorities because it eliminates the opportunities for bias to influence who makes the cut.
The point is not that there's a cabal of evil dudes scheming to exclude women. The point is that on a subconscious level many people stereotype each gender as having a specific set of roles. This biases their evaluation of the individual's performance.
An analogy: I have an alt account on youtube (to try to dodge Google's tracking) that has the gender set to female. When I'm logged into this account I get endless advertising for cleaning/laundry products, air fresheners, and diet solutions. This is despite the fact that my viewing habits are still those of a 30+ year old male developers and video game fan: coding talks, gaming stuff, and lets plays. Google's algorithms decided - based entirely on my set gender, and despite ample evidence to the contrary - that it should advertise stuff I obviously don't care about at me.
All I'd like you to consider that it's at least possible that the average hiring manager in a tech company carries the same kind of biased assumptions about what a female should be interested in and good at.
Such studies that show society treating women and men different based on expected gender roles has been done multiple times in many fields, and if we call that male and female cards then we must accept that for every place where a person can make a stereotypical assumption based on gender role then that is a place where one gender is favored and an other is disfavored. One prime example of those is with the justice system, where a male-sound name get found guilty more often and get harsher punishment because they don't get to automatic play the female card, as backup by blind testing at the subconscious level.
The average hiring manager has indeed some gender expectation. Based on some fascinating recent study done on judges, I would expect to see a major difference in how rational judgement a hiring manager do based on how long time it has been since last time they ate. The lower their blood sugar level are the more likely a person are to use shortcuts in their thinking.
Note that this has nothing to do with the belief of a "patriarchy", nor can it be translated into power cards that get played in some zero sum game.
I can honestly say I have never seen this mythical company, and if it existed then the wast majority of the bottom 10% of the population would no longer be male.