SGB - fully agree with your post about the two things to consider.
maria - I honestly think if you believe that institutions other than religious ones are not explicitly discriminatory, you're being awfully generous.
Explicit discrimination, in my book, isn't just about saying 'we fully believe women are inferior to men' (FWIW, relatively few religious institutions say this - they'll dance around it in all sorts of ways because they know it's quite blunt, even for their own followers).
Explicit discrimination can also be things like saying 'well, we would have more women doing career x, but women choose to have babies, so ...'.
I also think that by focussing on 'explicit discrimination' as somehow different and more important than other kinds, you end up (not deliberately, and 'you' as in people, not you personally!) downplaying the covert tactics, which are in some ways much nastier and harder to deal with.