I am sensing a general discounting of P effect. I caution against this. P effect is very real and significant. For example, imagine a variable pitch propeller, which varies in such a way as to allow the pitch to increase only on one side of the shaft and to decrease only on the other side. Essentially this is mechanically introduced P effect. This is known as cyclic pitch control, and is Igor Sikorsky’s great insight which made the modern helicopter possible.
You are exactly correct that cyclic pitch control made the helicopter a practical machine. If you check out the mechanics of cyclic pitch control, you will find that in order to control tilt of the rotor disc, the blade changes are introduced at ninety degrees to the desired reaction. In other words, for pitch, the blade changes occur as roll input causing the rotor to pitch and for roll control, the required input is pitch. When a helicopter moves too fast forward such that the retreating blade stalls, it doesn’t roll toward the retreating blade, but pitches up. As the cyclic pitch creates an asymmetric blade loading, the reaction is always at ninety degrees. In other words, the rotating blades of a helicopter and a propeller both react as any other gyroscope. Any force applied to the rim appears ninety degrees out of phase. P-factor is quite real, however it only affects pitch, not yaw.
When pitch angle increases, P-factor tends to increase pitch. This is opposite the stabilizing force from the tail and results in the longitudinal static stability being a little less power-on compared to power-off. But it doesn’t affect yaw.