I don't know if I follow your reasoning.
I think the only difference regarding the direction of the rotation of the gyro is that in one case one end of the gyro axis ends up pointing north and in the other case the opposite side.
If there is no gimbal to let the gyro rotate with respect to the boat, it will drag the boat with it.
If it's free to rotate in the horizontal direction, it will rotate with respect to the boat (stationary with respect to the earth) but the rate of rotation will depend on the spin.
In a commercial gyrocompass, it is fully gimballed, but it's maintained horizontally with weights (like a balanced arm scale).
You are right, Ta0, I was misunderstanding how the gyro compass works.
I found a better article about it and now I understand it a bit better;
the gyro compass points north, not because it precesses at the same speed of the rotation of Earth, but because it stops precessing when it points north.
I try again.
The small boat with the flywheel fixed horizontally in it, is a simplest kind of gyro compass.
It works
well onlybest at the equator.
In the drawing below, the small boat is at the equator, we see it from North Pole.
The flywheel spins, with its axis oriented east-west, (A), and it tries to maintain its orientation, but Earth rotates and in position B a torque from the small boat wanting to align itself to the sea makes the flywheel to precess, so the boat rotates on the plane of the water.
The boat rotates until the position C, then it stops rotating, and the flywheel remains with its axis oriented north-south.
The reason that the precession stops at this point is that now the flywheel axis is exactly parallel to the rotation axis of Earth, and, from here, the rotation of Earth does not change the situation of parallelism, so there is no more torque, and no more precession.
This works best at the equator because at the other latitudes
, especially near the poles, the axis of this flywheel, horizontal, would never be parallel to that of Earth, and the flywheel would tend to continue to precess, the leverage for making the flywheel to precess becomes progressively less efficient, until, at the North Pole, the gyro compass doesn't follow the rotation of Earth anymore.
Still the experiment with a small boat and a simple horizontal flywheel without gimbals maybe could be worth.
Depending on the direction of spinning and the orientation of the flywheel, before to point north, a clockwise or a counterclockwise rotation of the boat on the water could be seen, showing that Earth rotates.
