The gyrocompass orientation should, in principle, be obtainable by properly integrating gyroscope readings (and accelerometer? not sure if required).

You still need a way of determining the initial position/pose

I don't think so. The gyrocompass relies on gyroscopic precession. The precession itself can be obtained by integration, which gives orientation. Of course, the precision of this process may well be very low or unusable.

you don't need an initial heading measurement but you still need to know where you are

It would suffice to know the radius of the planet and it's rate of rotation.

Using only a compass and the knowledge that you were somewhere on the planet earth, could you determine the shortest route from your locations to Chicago?

Ah I thought you were referring to the gyrocompass functionality. Yes, for general navigation you need an initial position to determine your position from inertial measurement alone (as I said, you can determine orientation and latitude, but not longitude).

The laser gyroscopes do exist. They use lasers, plus the Sagnac effect, to measure rotations around on axis by interferometry.

https://en.wikipedia.org/wiki/Sagnac_effect