To the CFI in question w.r.t. to Gyro Precession, there is no mention of gyroscopic precession in the Airplane Flying Handbook in any part of the steep turns discussion (roll-in, established, or roll-out) so, if this CFI is teaching an Initial CFI candidate to teach his private students using material that is NOT included in any FAA publications, he should be able to provide a reference for where his info comes from (or a VERY VERY good argument that he might someday include in his OWN book lol). Regardless of personal feelings, I would not teach a pre-private student anything that does not have a citation somewhere in an FAA handbook or manual, but that’s just me.
With that being said, even though there is no mention of Gyroscopic Precession during steep turns, the physical principle of gyroscopic precession is still present.
Torque-Induced Precession (gyroscopic precession) occurs anytime a "torque" (rotational force) is applied to a rigid spinning body (ex. propeller).
So, to put it simply, GP happens anytime we have...
1. A rotational force and...
2. A spinning thing.
Does gyroscopic precession happen during a steep turn?
I will use only the "Established" Steep Turn example (NO rotation about the longitudinal axis ie. changing in bank angle. Bank angle is at a constant whatever angle)
In a steady state turn (steep or not so much), our direction (heading) is the only thing changing. Since our airplane is changing direction (heading), our airplane is rotating about the vertical (straight up - straight down) axis. When our airplane rotates about an axis, it applies a "torque" to any spinning things we have on board and since we have a giant spinning propeller on the front (or back lol) of our airplane, our airplane will be under the influence of gyroscopic procession during a turn (steep or not so much).
Steep Turn Induced Gyroscopic Precession is a "Control Pressure" problem, not a "Pitch Attitude" problem.
In any movement about any axis of flight in an airplane, we use a certain amount of “control pressure” to induce said movement. It is actually one of the most simple ideas in aviation (no 3 letter acronyms required lol), more control pressure = more movement. When we perform a steep turn, we are adding back-pressure on the yoke to account for the amount of vertical lift lost in such a high bank angle. Therefore, we can infer that if we had NOT SPINNING BODIES on our airplane, we would require the same amount of back elevator pressure, regardless of turn direction, during a steep turn. When we throw in a big spinning gyro on the front of our airplane we must factor in its affect on our back elevator pressure requirement.
Consider an airplane perfectly trimmed for straight and level flight at 90 kts. If someone else was to rotate the trim wheel forward without you knowing, how would you know? You would know because you would need to add back pressure in order to maintain the same pitch attitude thus remaining in straight and level flight. The same idea is true in a steep turn with regards to “felt” gyroscopic forces. When you rotate (change direction/heading) in a steep turn, you are going to feel a “nose-up force” or a “nose-down force” depending on which way you are turning (Left = Down, Right = Up). If left uncorrected, this gyroscopic force will cause the airplane to pitch down or up. In order to counteract this felt force in the control yoke, you are simply going to pull back more or less until you regain the pitch attitude you desire. It will require slightly more elevator-back pressure in a left turn and slightly less elevator back pressure in a right to maintain the level turning pitch attitude. That’s it. Your “SIGHT PICTURE” (what physical placement the nose of the airplane must be in in order to maintain a level turn) is not going to change DUE TO GYROSCOPIC PRECESSION because the only thing different in a left or right turn (WITH RESPECT TO GP) is the amount of control pressure required. All you will notice is a tendency for the airplane to pitch down or up and if you correct accordingly, your sight picture should remain unchanged. It is for this reason that, I believe, the FAA does not mention Gyroscopic Precession in the Steep Turn Discussion.
As for the reason why your sight picture DOES appear to be different in a left or right steep turn, I refer you to the FAA Airplane Flying Handbook (No gyroscopic precession required lol).