But it still doesn't tackle the possibility that our perceptions of what is required for certain jobs is false, and these perceptions skew the jobs that people are more inclined to go in to. The mechanics of each job have shifted over the decades, yet the same underlying personality associated with each job stays relatively rigid.
For example, a car mechanic. The skills and personality we'd associate with such a person would be what? Good with their hands, hard-working, willing to get a little dirty, some basic understanding of vehicles. How many of these skills are intrinsically tied to a specific gender? Women are biologically more dexterous, work ethic is pretty much even across genders. So what about the last two? Are women biologically less disposed to getting dirty than men? That seems pretty ridiculous to be honest. What's more likely is the societal pressure to remain "pristine" or "pretty" at all times for a woman makes them avoid the job. Understanding of vehicles though? What biologically would predispose either a male or a female to understanding a vehicle? Nothing, we haven't been around long enough to evolve any sort of biological influences over this. This is entirely society's perception that vehicles are a "manly" object, and thus encourages males over females. But if we're looking at this from a biological standpoint, a woman should be better qualified to be a car mechanic, and should have no biological disposition to avoid the job. It's almost all societal perception of the job and who should be doing it that shapes the fact that men dominate that job.
I feel like you're attributing too much to the individual, and not enough to the effects of societal pressure. While a person may be predisposed to certain activities because of their gender, this predisposition only matters if it effects their perception of what jobs are appropriate for them. Being empathetic or compassionate is a valuable trait in any field, but society emphasize the effects of it in very specific fields of medical and service industries. We pigeonhole traits to roles, when traits are not exclusively valuable in any given field. It's not only about opening up representation. If people don't want to work the job, they don't want to work the job. But it's about making sure that option is at least perceived as available. And frequently, society subscribes to the view that certain jobs are just better suited for men, despite no rational reason to support it. And it's these types of misplaced beliefs that need to be revised. While it's not a hard discrimination, such soft discrimination still influence decisions.