What makes Gerudo Secret Club so secret?
I finally have an answer! It's not just because they specialize in heat resistant armor for hylian men. It likely was a magic shop in a past life before it became a "clothes shop".
Two armor sets are available to buy at the Secret Club the Desert Voe Set and the Radiant Set. One is linked to the idea of male Gerudo. (although this set is made for Hylians to withstand the desert heat, nothing for any naturally born Gerudo men...) and the Radiant Set is linked to Dark Magic (necromancy) through its use of Luminous stones (contains souls of the dead) and the magic effects wearing the upgraded set gives you: Disguises you amongst Stal enemies and increases your attack power when using weapons made of bones. Basically using the souls of the dead to strengthen your power and join their ranks. In totk the Radiant Set is linked to a quest that has you cure Gloom Borne Illness. A villager's life force is actively being stolen from them by the Gloom so Link discovers the cure in Sundelions. The light power of the Sundelions exorcises the Gloom from a host. (Sonia must have absolutely treasured this flower's exorcism power as a Priestess.)
Combining the two concepts, (Gerudo Men and necromancy) paints a pretty good picture on why the Secret Club is so secret.
I like to think that generations ago, it was a secret club that a few Gerudo practiced Dark Magic in without the eyes of the law watching them. Perhaps, after the Imprisoning War, the few Gerudo survivors of the fallen villages all had to unify under Ardi as Chieftain regardless of their unique village traditions. Worship of the Legendary Male Born Every Hundred years (seems like it's own religion at this point) is outlawed because of his ability to cause disaster and mass casualties as shown in the war.
Oddly enough, after that law was established, its unclear if any more males were born to them.
Totk Masterworks says this is unclear two separate times. (Second time it's mentioned is in the section about the secret passage built into the castle over the seal.) It doesn't say anything about them not being born. Just that having a Gerudo King is outlawed because of their beliefs of impending disaster. Why would they automatically assume a male amongst them brings nothing but death?
"In terms of men since Ganondorf, there’s no remaining documents like that. It’s thought that the existence of boys was considered a threat since Ganondorf and they wouldn’t be able to ascend to the throne, though it’s ambiguous if there were any born in the first place. Were they no longer born, or no longer assumed to exist….? The reality of that ethnic characteristic is unclear."
This seems to imply males being born to them isn't natural at all to begin with. If it was natural, then they would continue to be born, but there wouldn't be a culture around arbitrarily crowning him a monarchal King. There might have been a more humble culture around him devoting his life to upholding light to make up for the evil Demon that used to fill his role. However the Gerudo do not seem to imply that is the case, nor is there any evidence that they murder any babies born to them.
There's also a passage in totk Masterworks that talks about Dark Magic being practiced by some Gerudo.
"Are puppets a secret skill of the Gerudo tribe?"
"Once, Ganondorf created a fake Princess Zelda even before obtaining the secret stone, and carried out spy activities inside Hyrule Castle. In other words, it’s clear that creating an imposter is not due to the Demon King’s power. If that’s the case, did it originally derive from a Gerudo tribe secret technique?"
"It’s said that the Gerudo tribe of the ancient past had the skill to use things like dubious animal bones and earth to control evil spirits. They would make something like a clay doll, and instil it with an evil spirit to become a puppet. Princess Zelda ridiculed the imitation of herself as Ganondorf’s ‘puppet’, and she was certainly correct. From the many acts of the ancient Gerudo tribe, they may have seen through this secret technique. Demon King Ganondorf may have used this technique in present day, while trying his best to control the consumption of his own dark power, perhaps he used it for reconnaissance."
Perhaps the answer is that the Legend of the 100 Year King is a sort of religious belief for an offshoot of the Gerudo people rather than a natural occurrence without magical intervention. The goal of this offshoot of "witches" is to bind the essence of a Demon King to flesh with their Dark Magic and legitimize his rule through disaster. We don't know why these witches exist but Koume and Kotake are certainly the leaders of this offshoot until they summon their "True King".
There's a lot in Masterworks to suggest the idea that controlling Gloom and Darkness is akin to necromancy. That's how the Blood Moon works to ressurrect monsters across the land and why Gloom is different from Malice. Malice is condensed hatred and negative emotions that arise like clouds from demonic spirits or cursed souls. Gloom is like liquid Demon essence searching for flesh to inhabit or steal life force and power from. Gloom is the vector in which Dark Magic is even possible. If this is all true then there really isnt any true or rightful Gerudo Kings at all. Just a small cult of Gerudo witches hellbent on installing a Demon Monarchy over not only the Desert Chieftains but all of Hyrule as their "True King".