Tywin's whole problem is that he's forced to rely on Cersei, and Cersei is a lethal combination of cruel and stupid. He needs her to pump out kids to solidify his hold on the Iron Throne, but he reasonably can't trust her to do anything but be a broodmare because, again, she's cruel and stupid.

In Westeros you can be cruel or you can be stupid, but you can't be both. If you're cruel you have to be smart, if you're stupid you have to be kind. Those are the winning combos, as we see. Tyrion can be cruel, but he's also smart. Jamie can be cruel, but he can also be kind. Tywin himself is cruel and smart. But Cersei's cruel and stupid and that undermines the entire Lannister effort.

Varys even talks about it at the end of Dance, how he specifically relies on her terrible job of ruling to pave the way for Young Griff.

Tywin's whole flaw was not having a smarter daughter.

>fat
>drunk
>demented
>perverted

kek, Cersei in Feast and Dance is a riot.