I'm going to approach this from a mythological perspective to show how the series strongly reinforces the idea of patriarchy via symbolism.
- Harry is an "Athenian" hero
Here I mean in the sense of the goddess Athena. This connection is explicit in that his familiar is an owl, the bird of Athena. Like Athena, Harry grew up without a mother. For Athena, this condition is held as one of her rationales for her tie-breaking vote at the trial of Orestes, where the son was acquitted for the murder of his mother. (One of the Apollo's arguments at the trial is that:
"The mother of what is called her child is not the parent, but the nurse of the newly-sown embryo. The one who mounts is the parent, whereas she, as a stranger for a stranger, preserves the young plant, if the god does not harm it. And I will show you proof of what I say: a father might exist without a mother. A witness is here at hand, the child of Olympian Zeus, who was not nursed in the darkness of a womb, and she is such a child as no goddess could give birth to."
This is one of the most misogynist statements in western literature, supports the modern anti-choice stance, and strongly reinforces the idea of patriarchy: the outcome of the trial is that the old powers of the Goddess (the Furies) are disempowered such that the new male hierarchy is affirmed, a reflection of the Olympian Zeus' usurpation of Gaia's powers (Titanomachy, Gigomachy).
The Metis/Athena myth is literally that unrestrained female wisdom threatens the established patriarchal order, such that Zeus tricks and swallows the Titan, giving birth to Athena from his head. Thus, wisdom/craft/skill is safely subjugated. (And, without Metis' intervention, Zeus never would have prevailed over Cronus.)
- Dumbledore is a retreat to paternalism
Essentially, the idea of comfort of having a father figure looking after one. There is no strong mother figure for Harry. (Mrs. Weasley does play a role as an ancillary mother figure, but, although she plays an important part in the Wizarding War, she is portrayed mainly as a traditional housewife. Note that the name Molly is almost certainly a reference to the magic plant moly, associated with the witch Circe in Greek mythology.)
Dumbledore is a "sky father", and a reflection Odin, the "first wizard". (The wizard's staff and pointed hat come from descriptions of Odin in his guise as wanderer, who undertook to various quests to gain magical powers. He was also the template for Gandalf.)
Norse mythology also reflects the usurpation of the feminine powers & attributes in that the Vanir, fertility gods of the earth, are subjugated and incorporated into the Aesir, often understood as skygods. Freya still has highest honor, but is subject to Odin's rule.
It's also been noted that Rowlings' take on the House Elves is a reprise of Kipling's White Man's Burden, which is as paternalistic as it gets.
Note that the chthonic wiki here is inaccurate in that chthonic powers are absolutely associated with Gaia, in that they literally exist within the earth, her body. Gaia gave birth to the Titans, Cyclopses & Giants, who were the nemeses of the Olympian order. Fertility goddesses such as Demeter are later additions, after the usurpation of female power, and function within the Olympian patriarchy.
Voldemort returns from the dead (underworld). He has a giant pet snake. [See Jörmungandr & Python]. The slaying of the Python is a prime example of usurpation of female power—in slaying it, Apollo becomes the god of prophecy. The priestesses are still women, but subject to Apollo. Similarly, most of the monsters in Harry potter are distinctly underworldly, as is mistrusted Slytherin in relation to the sky oriented, heroic Gryffindor.
This has the effect of reinforcing the idea that feminine powers are dark & evil, such that women can only be seen to acting in a socially productive way within the framework of patriarchy. (Athena as a prime example.)
Market Forces & Commercial Assumptions about Audience
It's no surprise that the central character of the series is male, as the general assumption until very recently has been that the fantasy audience is male dominated, and that even girls prefer male protagonists.
In fact, despite the international popularity of Pippi Longstocking, the OG "strong girl", one is hard pressed to find female protagonists, or even females with agency, in best-selling fantasy literature. (Notable exceptions are the Golden Compass, and Lucy Pevensie, who is the catalyst for the Narnia cycle. It is Lucy who opens the world for the Pevensie children via her own agency. Hunger Games is also a paragon, though technically science fiction.)
Hermione is a decidedly modern female fantasy character in that she has agency, and is an essential helper to the hero, but her role is subordinate to Harry's. There are many heroic, and villainous, women in the story, but all within the structure of patriarchy, whether good or evil. The Potter series at least is hopeful, casting the world as a fairly egalitarian society, where women have opportunities. To this end, Minerva, (the Roman name of Athena,) succeeds Dumbledore as headmaster of Hogwarts, even if the entrenched system is maintained.
It's notable that this condition surely influenced A Wizard of Earthsea, in that, when Le Guin was commissioned, there seemed to be no question the central character would be male. Le Guin did use this as a jumping off point to cast a female as the central character of the second novel, Tombs of Atuan, and eventually drove the series to explicate the history behind the disempowerment of women in Earthsea, which also functions as relevant social commentary. (One of the critiques of Atuan was that it was a female character within a male dominated hierarchy, but that was part of the point. Le Guin even rejected the characterization of Tenahu as "feminist", stating that she merely wanted to tell a woman's story involving womens' issues.)
The market significance of casting the Rey as the central character in the final Star Wars trilogy may be the upending of the trope that female protagonists in speculative fiction "don't sell".
