If you draw the relative wind vector using only the aircraft's velocity vector to represent the incoming airflow, I think you are overstating the P-factor.
The low-pressure field upstream of the propeller disc means that the air receives substantial acceleration before it reaches the plane of rotation. This has two components:
1. It enters the propeller plane at a higher speed than the IAS says. (In static runup, there is a positive incoming wind component, not a zero incoming wind)
2. The angle of attack, if any, is reduced by the above acceleration. (The stream tube is bent to nearly align with the prop axis)