How come some of the most important Greek gods were women (Athena, Artemis, Demeter) yet their society remained so strictly patriarchal?

You could answer this question really briefly: in all simplicity, gods were a category completely distinct from humans. Performing things that humans cannot perform is exactly the whole point of gods; it marks them as non-human. You wouldn't ask why didn't Greek men go around throwing lightning bolts just like Zeus, right? Well, because, rather obviously men cannot control lightning. Similarly, the Greeks would not think that just because Artemis hunts or Athena wears military gear, this is something that women should or could do as well.

As a simple way to illustrate this, the Hellenistic royals understood how these categories explained power and how they could be exploited. Once the Hellenistic kingdoms develop, the Hellenistic royals in Ptolemaic and Seleucid dynasties [at least, can't remember on the top of my head whether Attalids etc. practice this too] start marrying their sisters and other close relatives. Sibling marriage was absolutely considered incest among the common people, but gods did marry and produce offspring with their siblings: Cronus and Rhea, Hera and Zeus, Zeus and Demeter, and (important for the Ptolemaic monarchical ideology) the Egyptian gods Isis and Osiris. Not only does sibling marriage handily somewhat reduce the number of high risk title claimants, it could elevate the royals into a quasi-divine category above normal humans. Doing things humans cannot do makes one something more than just human, right? In the past, scholars used to think common Hellenistic Egyptians also started marrying their siblings to imitate their rulers, but in truth, if it ever happened, it must have been extremely rare and strictly frowned upon; the misconception is based on the fact that Greco-Egyptians used the affectionate terms "brother" and "sister" to refer to their spouses in letters, but it has nothing to do with their biological relationship. What normal mortal could be crazy enough to emulate kings and queens and think that they, too, could be divine?

So, that is the simple answer: gods and humans are two different categories of entities and thus their coded behaviours have nothing to do with each other. Gods in Greek religion do not work as moral exempla to be imitated, like e.g. monotheist saints and prophets - in fact, Greek gods do things that would be morally reprehensible for mortals all the time in the myths. And, what gods do or don't do in myths has very little to do with where their powers lie; Hera was the main protector of marriage, but what a marriage she herself had! And, Artemis was the most important goddess of childbirth, but she was emphatically a virgin goddess who herself certainly never gave birth.

Of course, we could go deeper into this question ("why is any one god woman or man, then?"), quite a few forests have been sacrificed to the question of gender and Greek gods, and historically it has been a very fertile ground for wacky theories drawing on Jungian and who-nows-what-psycho-anthropological-ism speculation.

This is the way I understand it. The Greek religion is a language without synonyms ("femininity" in a god is not the same as "femininity" in humans) and it has a grammar that is entirely its own. It really does not translate well to any modern mainstream religions (well maybe Hinduism or other polytheist religions, but my expertise ends here!), and it does not have a problem with something being particular and universal, or abstract and concrete, at the same time. E.g. the epiteths of Greek gods is a truly baffling thing: how is it that all these different embodiments with their own symbols, cult sites, rituals and functions are distinct "goddesses" on their own right - e.g. Artemis Agrotera, patron of hunters; Artemis Brauronia, protector of girls; Artemis Limenia, protector of harbours; Artemis Lokheia, protector of women in childbirth; Artemis of Ephesus, protector of Ephesus appearing in a rather... strange form - but at the same time they are all also "the same" goddess, Artemis? Jean-Pierre Vernant, a major scholar of Greek religion, reminds us that Greek gods are not "persons", they are "powers", and humans could simply appeal to particular, different aspects of that power when they wanted something from the goddess.

Gender - male or female god - could be what explained and signalled the divine "power" a Greek wanted to appeal to. It is, for example, very important that Demeter and her daughter, Kore/Persephone, are female; their daughter-mother relationship is not that different from human female relationships, and they were important patrons of different female life rites and rituals, but also their very "female" reactions to things explains somewhat metaphorically how universe works; winter comes because mother and daughter are violently separated from each other (Hades snatches Persephone as his bride), and Demeter's darker sides as the goddess associated with witchcraft and death are also part of her dark journey to the Underworld to retrieve her daughter and her revenge on the male gender. However, Demeter and Persephone are rather unique goddesses for being so "womanly"; notably, Hera and Aphrodite seem to completely lack any "normal" (lol I hope everyone realises here I am talking of only ancient Greek logic) female maternal instincts, although they have lots of children. And, of course, some goddesses were patrons of aspects of human society that were the very opposite of female, such as Athena as important goddess of warfare. I.e. with goddesses, being divine is always more important explanation for their powers and behaviour than being feminine, and as I already said, behaving divinely has nothing to do with behaving humanly.

Ancient Greeks wrote very little about the metaphysical nature of their gods, because belief was very non-important in Greek religion. The evidence is already rather late, but the writings of Hellenistic Stoics could illuminate a little bit how Greeks themselves might have explained the womanness of goddesses, if they ever stopped to think. Although, note that the Stoics believed that there was only one God, or Divine Rationality, that ordered the world, so they should not be taken as representative of all ancient Greeks. The Stoic philosopher Chrysippus explained that gods and goddesses simply act as metaphor for aspects of the divine and its different functions, and humans simply give those functions different names (Zeus, Aphrodite, Hera etc.). The sex of the god or goddess is therefore only the result of human mental operation, that links that particular divine function to either masculinity or femininity.

Does it matter at all, then, that for example, Artemis, often the goddess of supreme "masculine" functions (hunting), appeared to mortals as a virgin maiden and not as a man? Maybe, there could be two different ways to answer this question (which do not need to wholly exclude each other): 1) the origins or the 2) structure/symbolism of the cult:

The more boring explanation: the roots of major Greek gods go way back to the murky, distant beginnings of Indo-European culture and religion, and a female goddess of hunt might be a phenomenon that is thousands of years older than any "Greek" society, and therefore the gender of the goddess is simply a relic of ancient values and belief systems.
That Artemis is a woman, not a man as "should be", has explanatory force for the role of divinity in hunting. Perhaps it is important that the goddess of hunt is female, because hunting is so far removed from any female functions in the Greek society, that this paradox only emphasises and elevates the divinity of Artemis to the 1000th degree? Or; hunting was extremely important in ancient societies, but it is conceptually and ideologically difficult, because killing brings out the dark and chaotic sides of humanity. To hunt successfully like a human and not like a wild beast, one requires huge amount of skill and self-discipline. Maybe Artemis, as a beautiful and graceful girl who exhibits supreme, powerful self-control by always remaining virgin regardless of the fact that she is sometimes consumed by erotic desire (unlike her sister Athena, who is also a lifestyle virgin, Artemis is clearly a sexual being) and fiercely punishing those who do her wrong, dedicating herself fully to the forest, embodies more successfully the triumph of human mental strength over animal instincts than any regular male god of hunt could.

Anyway, you get my point, we could come up with dozens of different theories for 2) and at the end of the day, I am not sure if modern scholarship could ever come up with any one definitive answer, Greek religion quite efficiently escapes all definitiveness. Gender does matter in various ways in Greek religion, but Greek goddesses are a completely different category from mortal women, and therefore no matter how important or visible they were in public life, there is no reason why this would have had any implications for the status of their mortal sisters in Greek society.