Bad example with the nail salon thing. How many men have you seen with fingernail polish? There might be a few gay men interested in breathing those fumes all day, but they're likely employed in nail salons (or more likely own them). That field likely is almost-all-female naturally.
The child-care example is quite valid, however. Lots of men like being with small children, but our culture discourages that as they're seen as potential molesters. Another example is primary education: how many kindergarten or grade 1-4 teachers are male? There's simply no evidence that 95% of men have zero interest in being around small children, absent social stigma.
Anyway, I know you're joking about the cartel thing, but there's definitely a stigma about having men around small kids without female supervision, and this really is an inefficiency that's bad for society. Personally I think it results in a lot of women being around kids (because they fill the void basically) who really shouldn't be. As a society, we seem to have this idea that women are all wonderful, caring people when around small children, and it simply isn't true. A lot of them are nasty, heartless bitches and shouldn't be around any kids or have their own. Assuming men are generally bad and women are generally great with kids is bound to result in a bunch of these women taking jobs with children.