Yes. Those classmates didn't invent that sandwich stereotypes on their own, they got it from the environment.

I think the problem is not with men or women in particular, but society in general, as it has a concept that genders are socially different while they're [mostly] not. I strongly suspect, neither men, nor women are prerogative carriers of such stereotypes - although, naturally, as those stereotypes are discriminating to women, they're primarily opposed by them, not men.

Was there ever any doubt that the problem lies with men? Not for the last 80 years.