I also don't think Cersei is dumb or non-independent. I think most of her development is centered around her becoming free of associations with various men (Robert, Jamie, Tywin) . She's not the most skilled player on the board but she's not without chops.

I think Cersei's fatal flaw is her lack of humanity and her inability to gain allies that will be there if she ever falls into hard times.

One thing that is interesting about the female characters is that their weaknesses involve love choices, where men generally choose honor/duty. Honor/Duty seems to be a superior choice if you want to live in GoT, and then there are those who are conflicted, like Cersei.