Here: Why can I go either East or West from Toronto, and end up in Rome?

Regardless of the shape of the world, it would be because when you are traveling east or west you are actually traveling in a large circle with its center at the north pole.

On a spherical earth, a gyroscope would detect a semi-constant acceleration towards the ground.

Detect? All matter "detects" a semi-constant acceleration towards the ground in your view - doesn't it?

On a flat earth, a gyroscope would detect a semi-constant acceleration to the right or left, but none towards the ground.

The gyro (spin) - scope (see/view) was created for demonstrating the rotation (gyro) of the world - not its shape. If the world were flat, the gyroscope would behave exactly the same way it currently does and for the exact same reasons. Right?

Why do you think its function is dependent on the shape of the world?

What are the results of this experiment -- which is done every day via air-flight?

First of all, this is in no way an experiment - we are talking about mere observations. Secondly, the gyrocompass (and/or artificial horizon in an airplane) works - on that we can agree. Once again the question is why on earth you think that is dependent on the worlds shape. It isn't... and in the case of an airplane, which isn't even touching the ground, it is obviously doubly so.

I don't know, but I suspect that it's the former.

I'm not sure i'm fully understanding you. It seems you aren't properly understanding what the gyroscope is used for on airplanes. Maybe it will become more clear if you can answer some of the questions above.

In any case, your earnest engagement is appreciated!

Here: Why can I go either East or West from Toronto, and end up in Rome?

Regardless of the shape of the world, it would be because when you are traveling east or west you are actually traveling in a large circle with its center at the north pole.

On a spherical earth, a gyroscope would detect a semi-constant acceleration towards the ground.

Detect? All matter "detects" a semi-constant accelerations towards the ground in your view - doesn't it?

On a flat earth, a gyroscope would detect a semi-constant acceleration to the right or left, but none towards the ground.

The gyro (spin) - scope (see/view) was created for demonstrating the rotation (gyro) of the world - not its shape. If the world were flat, the gyroscope would behave exactly the same way it currently does and for the exact same reasons. Right?

Why do you think its function is dependent on the shape of the world?

What are the results of this experiment -- which is done every day via air-flight?

First of all, this is in no way an experiment - we are talking about mere observations. Secondly, the gyrocompass (and/or artificial horizon in an airplane) works - on that we can agree. Once again the question is why on earth you think that is dependent on the worlds shape. It isn't... and in the case of an airplane, which isn't even touching the ground, it is obviously doubly so.

I don't know, but I suspect that it's the former.

I'm not sure i'm fully understanding you. It seems you aren't properly understanding what the gyroscope is used for on airplanes. Maybe it will become more clear if you can answer some of the questions above.

In any case, your earnest engagement is appreciated!