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So if I held it solid and horizontal it wouldn't turn one way. If gravity is pulling down on the wheel and thus causing the precession, would a loaded bike on the ground not produce a force in the opposite direction through the axle upwards and make the wheel turn left? or is the whole thing cancelled by the fact that the wheel is held solid through the axle?
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is the whole thing cancelled by the fact that the wheel is held solid
The important thing is that the various loads (ground reaction, gravity, bearing support) net out to a zero torque. Any non-zero torque creates an acceleration, which also causes a precession when the object in question has angular momentum about an axis which is not aligned with the axis of the applied torque.
What you're seeing is precession, and the turning moment is driven by a torque trying to tilt the wheel, in the case of holding it in one hand driven by gravity.
When you hold one end of the axle, your grip is effectively a free pivot, so it doesn't react the torque generated by the mass being displaced from the fulcrum, so precession takes up the slack. On a Lefty, the joint between the axle and the leg is rigid, so it reacts the gravity driven torque and precession doesn't happen.