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.
Sorry if my comment ends up repeating what others have said in response to yours, but yeah, this chapter is pretty badly phrased. I really can't blame the translator either because this chapter is pretty much just one big exposition dump and would be challenging for even the most fluent translator, and the queen seems like she's rushing through the explanation because she thinks Leon is completely comprehending the deeper meanings behind each part of the plan and how it has unfolded.
Some of this is directly said here or in past chapters, but some of this is my speculation so take it with a grain of salt.
From what I've understood, the original political landscape was that the world was a kind of free for all where relatively strong adventurers would claim floating islands, establish themselves as feudal lords, quickly amass a small army because of how cheap and easy it is to use floating crystals from the islands to make airships, conquer other lords to amass even more power, and finally they'd attack the kingdom, the biggest feudal family of the time. They were always defeated thanks to the royal ship, but they always cost the kingdom significant casualties and resources. So it's pretty much exactly what just happened with the principality.
Because of the misogyny of the time, these households were primarily run by men, both in terms of military command and training and in financial terms. The kingdom's royal family wanted to both prevent these attacks that could come from anywhere AND slowly erode at the feudal lords' power until they could consolidate all the power within the capital. Training commoners won't be to give commoners power but just managerial positions in a centralized government where the royal family holds all the actual power. With a centralized government, warlords can't build an army and conquer neighboring lords to amass an even bigger army before attacking the kingdom because there would be no other feudal lords to conquer.
Simply enacting policies that transferred power from the men in these families to the women wouldn't do anything, they could just as easily train the women to be military and financial leaders instead. Even if physiological differences had an effect, it's pretty minimal in a world were there's abundant magic and cheap, mass producible airships, and the royal family knows this. The REAL crux of the plan was to simultaneously create the centralized academy in the kingdom's capital, and by playing to the feudal lords' egos and need to be seen as legitimate nobility, lure them into sending their female children to the prestigious academy thinking they were going to get married to powerful men and form powerful blood ties only to be brainwashed into living a life of indulgence and hedonism where they spent all of their money on fashion and parties because all of the women already living in the kingdom were living like that. When in Rome, do as the romans do. Any children they sent, male or female, also saw no point in attacking the kingdom directly because they can see firsthand how powerful the royal family is whenever a warlord attacks and gets wiped out by the royal ship. It got out of control tho and turned into legitimate misandry and the popularization of male sex slaves, but the plan effectively stopped most lords from attacking by draining their funds so the problems were ignored.
The same thing probably would have worked if they encouraged hedonism among the male children of the feudal lords coming to the academy, but it was faster to enact because of the dwindling number of male successors from the constant fighting and because those male successors were still ambitious and would be harder to peer pressure into spending their time and money that way. They may be deterred from directly attacking the kingdom, but they would still try to gain political and financial power inside the kingdom. The most powerful families caught on quick and raised their daughters to not fall into the trap, but the abundant smaller feudal households didn't have the same insight or information networks.
In short, it's not that the royal family thinks the feudal families are weaker being run by women, it's that they created a culture of caution of their military might and hedonism fueled spending that leeched away the feudal families' funds, and women lacking training from their families were an easy target.
It still makes no sense tho because feudal lord's aren't just going to go belly up for the kingdom's policies that favor women and hand over control of the family finances to their daughters. Especially if they're misogynistic at the start of this plan, they would have no problem disowning their daughters when they start wastefully spending their family money.