It's clear that P-factor is entirely different from gyroscopic precession, except that their forces are transmitted to the airplane through the propeller shaft.
Not so. The force generated by P-factor is the same as any other force acting on the rotating propeller (gyroscope). Precession is simply the fact that
any force that is applied to change the axis of rotation acts at ninety degrees from the applied force. It doesn’t matter if the force is generated aerodynamically or applied through the shaft. Precession happens. P-factor exists but it produces a pitching moment not a yaw.