The one place that I agree with you on this however, is childcare. I've dealt with a large amount of bullshit while staying at home with my newborn, and male caregivers are typically assumed to be incompetent.
Did you friends mention how children to men who take up female professions get teased in school because the job choice that their dad took? How their spouses don't want to tell friends what their partners job title is. How male nurses get question like "are you not a man, why are you a nurse?". People who walk up to a male midwife and accuse him of being a pedophile, ready to punch him? Read any of the multiple news article on how male preschool teachers do not feel safe at the job, feeling like they are under constant accusation by parents and co-workers? Same articles that talk how they dislike being the "representative" male, always chosen to do "male" stereotypical job activities?
But more importantly, the data don't lie. Male teachers and male students that are studying to become a teacher exit the job profession in rate about 450% higher than a female counterparts. Are we going to use the same old arguments that its something genetic that a male teachers starts working and after a year or so decide leave the profession?
The next step should be to ask why women working in a female dominated profession are harassing the men in the same profession, or why men working in male dominated professions are harassing women. Maybe if we could figure out a good answer to that, we could turn the tide of more work professions becoming gendered with unbalanced ratios.