Bob Woodward chronicled the access Rice had to Bush in his book "Bush at War"[0]. His impression matches your own. According to Woodward's account, her de facto role as National Security Advisor (before her tenure as Secretary of State) was to meet with the principals (Rumsfeld, Cheney, Powell) without the President, and then discuss alone with the President the positions of his various cabinet members. Rice met one-on-one with Bush much more often than the others, especially Powell. That said, her explicit role at the time was more about collating data and balancing priorities, not advancing her agenda. But it is definitely fair to say she had the President's ear during this period, perhaps more so than any other national security administration official.
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.
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...