The increased left yaw when the tail is raised is caused only by gyroscopic precession.
Yes, but I'm wary of using absolutes such as "only" which can come back to bite you.
The yaw due to asymmetric thrust reduces as the tail is raised, it does not increase.
Why would it change at all, the relationship of the propeller to the aeroplane remains the same? Of course, the propeller slipstream stretches and the yawing force from this will therefore reduce with the airspeed increasing but it will always be present. At the slow take-off speed the slipstream yawing force will still be substantial and require a sustained application of opposite rudder.