So: The Eumenides isn't going to make it into a list of Top 100 Most Feminist Texts ever written any time soon. In fact, if it did make its way onto a Women's' Studies syllabus, it would probably be there as an example of how bad misogyny can get.
No matter how bad you think "bad" can be, The Eumenides is worse. In this world, you're justified in killing your mother out of revenge for your mother killing your father (family life in The Eumenides is messed up to say the least), but you're not justified in killing your husband out of revenge for your husband killing your daughter. Paper trumps rock, and justifiable matricide trumps justifiable mariticide.
Yeah, that's bad—but it gets worse. Apollo gives a passionate, Atticus-Finch-if-Atticus-Finch-was-evil-instead-of-awesome speech in the defense of his client's killing of his mother:
(Apollo): "[…]. The so-called mother is no parent of a child, but nurturer of a newly seeded embryo; the parent is the one who mounts her, while she conserves the child like a stranger for a stranger, for those fathers not thwarted by god. […]" (657-666)
There you have it, folks: the mom is not related to her kid. We repeat: the mom is not related to her kid. Wow. Just: wow.
But before you start throwing your copy of the Oresteia trilogy across the room, we want to move things into the realm of the symbolic. Apollo wasn't anti-mama. He was just conforming to every stereotype of the scheming lawyer and making a clever argument… which was actually in the name of a greater good. Was it sexist? Oh, yeah. But it served a point.
Because, see, Apollo and Athena's whole end game in this trial is to move social order away from revenge and towards a judicial system. In abstract terms, he was trying to move justice away from the realm of the heart (need revenge now) and towards the realm of the mind (hmm, let's weigh the pros and cons). And the idea of fatherhood, at least back in the day, was much more associated with the realm of the head than the heart.
If you're a new dad in ancient Greece, a few things are going through your head. The first, obviously, is "Squee! Time to buy all the baby-sized togas!" But the second is "Okay, let's think rationally. Is this kid really mine? Okay, he has my eyes—check. He has my weird toes—check. Okay, cool. Time to break out the wine-skins in celebration!"
Mothers—not so much. They're basically tethered to the child (no actual argument can be made for the fact that it isn't theirs) and are probably having all the feels as a result of a) happiness b) hormones and c) giving birth in the era before epidurals.
So this "moms aren't related to their babies" argument is an elaborate metaphor for the idea of revenge, and justice. The feeling of revenge can be likened to a mother's claim on her kiddo: if a kid comes out of a woman's birth canal, it's hers, just like if you feel a need for revenge, it's yours. There's also the implication of overwhelming (traditionally "feminine") high emotion.
Good justice, on the other hand, is full of judgment. You have to take a detached look at the situation. "Is this kid mine?" requires the same thought processes, necessarily removed from emotion, as "Is (insert act here) justifiable?"
So The Eumenides isn't just being a sexist pig. It's also being a very clever, metaphor-using, abstract-thinking pig.