Yy to the causal link being back to front - employers get away with paying less in roles that are traditionally thought of as female. There's nothing about those jobs which makes them worth a lower pay packet.
Also, same job different pay - I suspect that this is far more prevalent than many realise but that it isn't widely publicised because of gagging clauses in settlements. In the past I had to take my then employer tocourt (being ddeliberately vague here because of just such a gagging clause). I got a hefty pay rise and back pay. It was part of a group action brought by my trade union at the time.
It would be next to impossible to fight now. In the last year May's government has changed the law (not well publicised - I didn't see anything in the press). Trade unions can no longer bring group actions but have to fight each case individually - so instead of funding one case and being able to say "look how many individuals are affected" they'd have to fight scores of cases separately, making it both financially untenable for the union's legal fund and obscuring in court the fact that the discrimination was affecting lots of women at the same time, it wasn't just a one off cock up.