IIRC a simple SAS takes the displacement information from the gyro, integrates it giving you displacement over time = rate and then opposes that rate using a feedback loop to modify the output.
I think you are mixing two concepts together. A SAS would normally use a rate gyro which is giving the rate signal directly. A secondary input could be from an attitude gyro signal which the SAS computer
differentiates to get rate (dθ/dt).
An ATT system would normally use an attitude gyro signal directly, but could use the rate gyro signal as a secondary input, which it then
integrates to get attitude.