A fairly obvious flaw in the grand design of the Bene Gesserit

Frank Herbert's masterful Dune offers a science-ficton saga unlike any other. Not only is the story intelligent, gripping and thematically interesting, the sheer depth of the world-building is staggering. I have loved it since began reading the first book at an impressionable young age, and I can talk about it at length. What I want to talk about right now, however, is what I perceive to be a logical flaw in the underlying plot. This flaw can be attributed to an oversight on the author's part, or can be explained as an error committed "in-universe", as it were. Before I go into it, be advised that I'll be spoiling some big things here. If you haven't read Dune, go do that. Read the first book, and then come back here. (And talk to me about it.)

So, you've read it? We're set? Okay, here it is: the Bene Gesserit want the Kwisatz Haderach to be born, and they want him to be under their exclusive influence from the outset. To this end, Jessica was explicitly ordered to give Leto only daughters, no sons. This was presumably because Leto was so very popular in the Landsraad. Any son of his would inherit his position and standing, would be raised well, and would most assuredly be at least a near-Kwisatz Haderach (and would thus possess astounding potential no matter what). As support for House Corrino continued to dwindle (in part due to the machinations of the Bene Gesserit themselves), a son of Duke Leto could become a viable contender for the throne.

In other words: such a son could be a critical threat to the Bene Gesserit  plans, which aimed at putting their own, controlled Kwisatz Haderach on the throne. They had themselves been instrumental in eroding support for Shaddam IV, and in ensuring he only had daughters. Presumably, in their plan, Irulan would briefly become Empress or Regent, while marrying a consort that the Bene Gesserit would carefully select. She would presumably have a daughter, who would be married to the intended Kwisatz Haderach, as yet another way to control him (since she'd be to him what a Bene Gesserit wife is to every high nobleman: the power behind his throne).

Of course, Jessica disobeyed her instructions and gave birth to a son, who indeed became the very thing the Bene Gesserit had so feared. My question is: why did they allow it to get to that point? Given the premise that Jessica has given birth to a son, surely it would be of the very greatest ease for the Bene Gesserit to fit him right into their plans? As it transpired, sending him to Arrakis was extremely dangerous. It's a planet where the prevision-enhancing Spice is literally in the air and in the food. It stands to reason that his potential would bloom there! And the Bene Gesserit knew of the plot against Duke Leto! Surely they must also have grasped that this could easily turn Paul against them, if he survived (as he indeed did)?

All this insane risk, when a much more obvious solution was available: simply instruct Jessica to have a daughter after all. This is something she ended up doing anyway, also to very dangerous effect precisely because she was on Arrakis by that point. But if the Bene Gesserit had simply made it clear that Jessica, who was in a precarious position because of her disobedience, could make things right by producing the sought-after daughter after all, then Paul would presumably have had a sister much sooner. A sister who could then be trained as a Bene Gesserit. Whereupon she could be dispatched to seduce Feyd-Rautha, thus becoming the mother of the intended Kwisatz Haderach, who could subsequently be raised in a Bene Gesserit-controlled environment.

Meanwhile, the solution in regards to Paul is obvious: just manipulate events so that he ends up married to Irulan. Shaddam IV himself regretted that he couldn't have married his daughter off to Duke Leto. Why not marry her off to Duke Leto's son? Sure, Irulan is older than he is, but the Spice has its youth-extending effect anyway, and we know that Paul did actually end up marrying Irulan. Clearly, the match is a realistic notion. Leto and his backers in the Landsraad would furthermore perceive this as getting exactly what they wanted. Irulan could simply give her consort, Paul, a daughter... who could subsequently be trained as a Bene Gesserit and married off to the Bene Gesserit-controlled Kwisatz Haderach. Since the latter's parentage would be secret anyway, nobody would know that it was an incestuous union (and the Bene Gesserit had already demonstrated not to shirk from that in the service of their plans).

With that, the Golden Lion Throne would have come under the direct control of the Bene Gesserit. Their goal would have been achieved without all the insane risks they actually ended up taking, and nobody would even know that there had just been a huge coup! Naturally, this would make for a less spectacular novel. And that, presumably, was the real reason why it didn't pan out in this manner. But in-universe, we can only state that the Bene Gesserit made a very obvious mistake. It cost them dearly.