You attempted to dress up quotes from a Cosmopolitan news paper as it was a study based on data. It don't work, and I am not sure why you trying such a futile attempt at misdirection. That is anti-science, and is fatally doomed to fail at persuasion.
It clear that you won't admit that consumer preference exist, as it would shows a clear link between gender specific incentives and behavior. You can try to play ignorant, but it won't change the data. Incentives has a real impact on human behavior, and it is trivial fact in psychology.
Finally you admit that men, just maybe, just possible, just as a theory perhaps, might face a unique pressure to make money. Funny that. And yes there is a strong possibility that gender segregation has a multitude of factors (a fact that was well written in a government issued report on gender segregation in the field of education). The problem statement then changes at this point to define which factor has the bigger impact, and we get back again to look at experimental data to see what has a strong impact and what has a small impact.
Let for example guess there exist a study on what jobs children want to do when they grow up. What will the average income of those jobs be when comparing boys vs girls preferences?
Or let say a study at a bit later time in life, and look at when people enter the work force. Will we see men entering the work force earlier and with greater haste than women, and what incentives is there for young unmarried men to get an income around the ages of 18-25?
We can look at studies that survey what teens want in a job, and how they select which profession to study. We can continue by looking at whom and what impacts behavior during this crucial time period in a person life.
We can also look at the primary cause of gender segregation as highlighted in that government report I mentioned earlier. Do people feel more confident when entering a group that they belong? do they feel more self-doubt when they are a local minority (such as in a class room or work place). Does ingroup and outgroup exist as theories, what is the scientific proof for it, and does herd behavior exist in human psychology.
I will note that in the government report, no where does it even entertain the idea that women are crowding out men from the education system. Similar to bloomberg it does record a historical change where men used to be the dominating gender in education, but is no longer. Instead it gives other explanations, such as the government cutting funds and thus the social status of teachers dropped. When money disperse from a profession it also caused the gender ratio to favor more women. When money increase, the opposite happens. Fairly simple statement to prove or disprove by data. Here is your golden opportunity: convince someone on the Internet of your theory by disproving that a influx of money has an impact on gender segregation.