I actually found this explanation pretty interesting and reasonable, even if it's not perfect, so I'll give a shot at explaining it.
That was too many words for something so contrived. Is there any TL;DR?
Basically, some time ago the Kingdom had a serious issues with Barons/Lords of their own nation either withdrawing or full on attacking the capital. This was due to how easy it was for those in possession of flying crystals to make airships. Being cheap to build and maintain, anyone who had access to the crystals suddenly had a massive spike in their military power. The Kingdom, being in a static location, could only react to said Barons/Lords when they were attacked, putting them at a disadvantage since they couldn't exactly pre-emptively shut down
every Baron/Lord.
They then came up with a plan to slowly weaken these Barons/Lords firstly by shifting the power structure towards women (I would assume most, if not all of the Barons/Lords were men) and secondly by creating the academy to accrue talent + insulate power and influence. If all the elite's children are at the Kingdom, it makes it harder to attack for a number of reasons and the Kingdom becomes more powerful to boot. It's a win-win for them.
The ultimate goal of this was to slowly weaken the Baronies and Nobility over time, then replace them with trained + educated commoners raised under the ideals of the Kingdom. Any Nobles that were worth their salt could remain, and the rest would be purged.
This plan got twisted along the way, leading to the state the Kingdom is in now.
I'm no feminist, but this reasoning they have for weakening the nobles is incredibly sexist and rather offensive. It basically hinges on women being less capable at politics and power plays than men (which is flat out wrong) and how this would cause the power of the noble class to diminish. Maybe in wartime, yes, as it is still the men who fight in wars and hold the military power. But in peacetime, this makes zero sense.
...or I'm totally misunderstanding something, because sadly the chapter's explanation was incredibly badly phrased and I could only gather the above from the comments here.
I don't think that was the intention in my opinion. I think that in the setting the positions with the most power (specifically military) were almost exclusively held by men, and the Kingdom had a problem with these men wreaking chaos and havoc internally while the Kingdom still had outside enemies who really,
really don't like them. The shift of "power" from men to women was one of the methods they used to try and consolidate everything until they could "fix" said problem at a later date. It wasn't supposed to get this out of control.
You may have a point relating to the peacetime political atmosphere, but we haven't really gotten that much in the way of how that's worked in the story since this is a setting that's intentionally more outlandish in premise.
It's definitely all a bit silly and fantastical, but I don't think the intent was "men are too competent, so let's give the power to those stupid women instead haw haw". At least that's how I look at it.