She nevertheless, imo, played a pretty big role in paving the way for an attack on Iraq and was rewarded for that with elevation to cabinet afterwards.
Do you have any reason to believe that she was in the dark in 2003?
On an semi-related note, "Bush at War" is a great, quick read about the months following 9/11, with a breathless day-by-day account of the meetings and phone calls held between the President and his advisors. Woodward had unparalleled access during this period, including multiple in-person interviews with Bush both in Washington and in Crawford. Fair warning, he does not mince words on his dissatisfaction with Bush's performance.
[0]: https://www.amazon.com/Bush-at-War-Bob-Woodward/dp/074324461...
Sure, if the claim had been "had Bush's ear" rather than "was a senior member of Bush's cabinet", I wouldn't have said it was factually incorrect.
Can something be "common knowledge" and still be false?
While the person involved later became a senior cabinet member as Secretary of State, at the time the war in Iraq was launched, Rice was not only not a "senior member of Bush's cabinet", but she wasn't a member of the cabinet at all.
At that time, she was Assistant to the President for National Security Affairs (popularly, "National Security Advisor"), a position that is a member of the National Security Council, but not the cabinet.