My other comment adds more context to this. But basically no, I was just trying to say that:

1. Stereotypes are harmful because they pigeonholed people and attribute qualities to each individual, however they are also grounded on some level of truth, which provides some level of utility. Dismissing them is dismissing that element of truth. Be honest, if you had a small child and had to leave them with a random woman or a random man for a few hours, which would you be more comfortable with? Of course, if you had to leave them with someone you knew well(like your brother vs your sister) there would be almost no difference.
2. Women and men (talking about group averages, not individuals), have different predispositions. In US 97.6% of preschool and kindergarten teachers are female. Why do you think that is?In my view, it's a combination three main factors: ability to do the job, finding the job fulfilling and discrimination. Even if we got forcibly get rid of discrimination by enforcing 50/50 quotas, why would we want to force people in roles that they would be worse at and find less fulfilling?

It does suck for the people who actually have the talent for jobs in fields where they face discrimination and we should always be looking for sensible ways to approach discrimination. I will note however that there is less discrimination now that in any moment in history, yet people still try to suggest the most radical of changes, which annoys and baffles me to no end.

So let's take testosterone for example, it is a biological fact that the average man produces more testosterone than an average woman. Testosterone is linked with aggression. Would you rather leave your kids with someone who is more likely to be aggressive or someone who is more likely to be empathetic?

So you want to force men into jobs they don't want to work and force women in jobs they don't want to work?

No, I think it's more complicated than that. Sociological forces exist, but so do other factors. There is the greater male variability hypothesis for example. There is also evidence to suggest that in general women tend to gravitate to jobs that deal with people and men gravitate towards jobs that deal with things. There is also the gender equality paradox - as countries become more egalitarian (like Scandinavian European countries for example), the gender gap actually increases.

I am not saying any group is better than any other group, and I am not trying to pigeonhold anyone either - the in group variance is actually higher than the variance between groups, so there are women qualified for any position. Equality of opportunity should be one of the main goals of our society, however expecting an even 50/50 split in anything is ridiculous.

How does this privilege manifest at lower levels of society? Would you support 50:50 quotas for garbage men, plumbers, builders, mine shaft workers?

I think you overestimate how much privilege matters. If person A can do the same job at half the cost that person B can, in a capitalist society person A will get hired significantly more often.