Hello! Student of history here who just finished three classes on Greco-Roman myth and three Attic Greek language courses. This will be a brief but informative answer and if there are any follow up questions I will be happy to answer them!

Part 1

Let's start with the first part of your question, "How were the Ancient Greeks so misogynistic and patriarchal". Well that's a complicated question and many layers to this answer but I will go over some of the larger issues in Greek society. First off in the ancient world, preserving the family lineage was of utmost importance. Greek men wanted to know that the children their wives had were their own. (It was thought that the man's sperm was the main ingredient in making children). In Homer's Iliad Book 2 the so called "Catalog of ships" can be seen as an example of how the Greek mind took lineage seriously. Despite being a slog to read through, this information was important enough to preserved. Female sexuality made Greek men anxious and one way of preserving the integrity of a family lineage before the advent of DNA tests (sorry no ancient Maury Povich) was to strictly regulate the behavior of women. This meant that men could adventure and engage in extra-martial sex while women were restricted and relegated to the domestic sphere. To go back to Homer a key example of this marital dynamic appears in the Odyssey as Penelope must fend off dozens of potential suitors two decades after her husband left for war while Odysseus himself engages in multiple affairs during his travel. The implication is that Penelope must remain faithful to her husband to protect her honor but Odysseus has no such constraint.

Another element of Greek society that enforced unequal gender performance was the warrior culture of the Greeks. Greek cities were in a state of near constant war and as men were the only citizens eligible for military service, the idea formed that they were entitled to more rights as protectors of the city. (This thought process is critiqued in Euripides' Medea when Medea comments that she would rather brave the front line 3 times than give birth once.)

I would like to take a moment to note that when we discuss "The Greeks" there really was no single unified entity. Each city-state or polis operated based on their own rules and traditions. An example of this can be seen in Athens treatment of women vs Sparta. Spartan women gained more autonomy than Athenian women since they were required to take care of the city while their men were away at war. They trained and exercised to strengthen the polis as a whole. Athenian men would have seen such practices as uncouth.

Part 2

"worshipping powerful goddesses like Hera, Athena, Aphrodite, Artemis etc."

When it comes to the immortals, they followed different rules than us mortals. An example, Hera and Zeus were siblings and married but such incestuous marriages were not permissible in Greek society. (See Oedipus Cycle for how the Greeks felt about incest). Egyptian royalty and the Ptolemies who controlled Egypt after Alexander's conquest practiced royal incest but that was not the norm. (A practical motivator for the Ptolemies was the anxiety of impure lineages like we discussed earlier.)

While male Greek gods could be emulated by humans, (Think Jason in the Argonautica emulating Apollo) the behavior of female Greek gods could not. Artemis and Athena were perpetual virgins and since the Greeks were focused on creating progeny, a woman who did not have children with her husband would be looked on with scorn. Athena and Artemis also wielded weapons, yet women could not serve in the military and so could not emulate their behavior. Aphrodite was a sexually powerful goddess but the Greeks feared sexual power in their women since it threatened their lineages. These goddesses were ideals that simply could not be imitated based on the structure of society. Hera was the closest a woman could come to acting like a goddess since Hera was associated with family and marriage.

When first studying Greek mythology Athena stands out as a counter to all this Greek misogyny, but alas no comfort can be found in her character. Athena being born from Zeus's head aligns closer with her father, not her mother who was swallowed by Zeus. This gives Athena a distinct masculine quality and mindset. Athena helps many male Greek heroes such as Odysseus and Heracles with their trials but Athena punishes women who act out of place. An example of this behavior is Arachne who is turned into a spider by Athena. She enforces patriarchy through her actions and despite her outward appearance, thinks and behaves like a male god.

This has been a rough synthesis of my last 2 quarters coursework. For any claims I have made I will gladly provide a citation upon request.