I think you are conflating some issues.
Competitiveness is not the same as “ability to compete”. Competitiveness is the “urge to compete” and it involves several more basic concerns combined into an aberrant end effect.
The female much more strongly feels the urge to compete within the female arena. Jealousy (the urge to fight for ownership) among women is much stronger than among men relative their own concerns (despite what they try to tell you). Such things happen for reasons. They are not just arbitrary attributes that can be chosen to be different.
The male was designed (by whatever means) to be less concerned over immediate threats, bumps, and bruises so as to be able to pursue a target (food, enemy, female,…). The female was designed to be more concerned with immediate threat so as to protect the offspring, sometimes still within her. The female was naturally designed to be more emotional and persuadable, without which there could never have been offspring. That attribute lends itself greatly to when the female will instinctively yield and in which type of competitions she will naturally excel (or fail).
Thus when an immediate light threat is sensed, the female immediately notices but doesn’t necessarily see the distant threat or goal. The male is the other way around, often missing the immediate and obvious as he instinctively focuses on the distant and perhaps far removed concerns. This distinction is far more relevant to why men are usually the natural leaders. Of course, nothing on Earth is natural any more as Man attempts to gain control over all life and bend it to his will.
The few times when women have become the standard for leadership in a society, it has been due to divisive medication of the males and such gets discovered and dismissed eventually.
Today, the physiology and mental prowess of the male and female is not exactly what nature produced, but what Man is designing them to be, so all kinds of confusions come about as specialties are lost to a gray race and gender artificial design.
The urge to compete, competitiveness, is aimed to where media, medical environment, schooling, and laws guide it to be. Although the natural design has not been entirely removed thus the male and female still show distinction between which kind of situations will bring out the competitive urgings and within which they will be more capable competitors. As hard as Man tries, the male and female are still not equal in many regards.
The jungle environment presumed to be from whence the design of humanity sprang, requires the specialty of abilities and thus one gender must be better at some things than the other. It was the teamwork that allowed the human family to overcome the competition between species.
Generally after the males have conquered the foe, established the empire, or built the house, the female takes over to “doll it up”, make it more comfortable and rodent free, that is her specialty from “nesting”. Note that throughout history, the Queen is running the kingdom very shortly after the kingdom conquered its last foe. This is a pattern dating back to antiquity.