>>283499849
By that point dealing with Ougi was absolutely a priority, all of Gaen's current and former network of associates had been committed against her and Ougi had managed to deflect them all. Gaen had multiple contingencies and came out ahead of course, but she had to stick her own neck out there between Kiss-shot's claws to do it. And even then she didn't get everything she wanted, since Araragi and Oshino save Ougi instead of letting her be erased.
This is all mostly just going off my memory of Final Season of course, and Gaen's exposition dumps are pretty dense, I'm sure there were even more layers to her keikaku. Like I don't remember if Tadatsuru actually got played by Ougi or if it was all part of Gaen's plan, or if he got played by both of them.