Picking Up Chicks: Animal Style

So far, in every chapter, Dawkins has discussed the behavior of only males in the different species. Sure, he mentions females and how many eggs they lay and so forth, but it seems to me like the males are doing everything in their power to get the women. They fight over territory, they try to get a high "social" rank. Males, as Dawkins describes, are always competing with each other to see who mates.
Sure, females do begin to play a role after they have had kids. Building the nest. Hunting. Feeding the young. However, what do single females do during mating season? My understanding so far is that they just sit there waiting for the winner of this eternal struggle between males to be the ones who reproduce.
Is nature sexist? In every species the man is constantly fighting for the woman and doing everything in his power to be the one that gets to reproduce. Sexism in humans is different because of the whole voting thing and equal job opportunities, etc, but the same basic principle of this sexism is there in all species. The female is chilling while the males are breaking their balls trying to win out. Also, the repercussions of losing the fight for the female are disastrous. As Dawkins describes, males who don't breed soon die of starvation. How come everything developed this way? Why, since the beginning of species, has the man been the one to try and win the woman?