Absolutely! That’s what I had in mind, and even the Starks. (Which is interesting considering that Catelyn was raised with a lot of knowledge about politics and running a household, because she was heir to House Tully until Edmure was born—a good portion of her life.)
Honestly, no idea if she did! I get the sense Catelyn might have just because of how Hoster was as a father, and that he seems more ambitious than Ned. I wouldn’t be surprised if Ned wanted to shield his children more from the politics or rest of Westeros as much as he could due to his trauma.
I’m not super sure if George thought everything through but things get more fleshed out as he writes worldbuilding stuff (that’s not uh TWOW). Like, I think if anything the isolation of the Stark girls and Cersei are exceptions rather than what should have been or what’s politically advantageous. The Reach does well in cultivating alliances by having a women’s court of a sort around Margaery. We see that for Rhaena (first one) and Alysanne, as well, and it seems like something Alyssa made sure they had from a young age. But this could also be due to George’s gardening.