In a steady state, straight climb, gyroscopic precession is not acting.
There is where we disagree. In a steady state climb there is P factor yawing the plane to the left. That is a steady force applied to the propeller at the 3:00 position. Due to gyroscopic procession it has to result in a displaced nose up force acting throught the propshaft at the 6:00 position. The plane of the prop isnt changing but the forces are still there.