Very sophisticated gyro compasses use this effect to find true north but most that we would use just function to sense rate of change. The gyro and the compass work together with the compass determining the heading and the gyro sensing changes in direction. Gyros are offered as an option to sense rate of change. With this information, the autopilot can better determine how much rudder to apply. This is helpful down wind or in following seas when the boat tends to oscillate in yaw.
Precession causes the axis to drift slowly.
If the fluxgate is maintaining overall heading , this is not a problem. The autopilot is looking at the gyro to determine the short term rate of change not the heading . I suppose it would be of some use up wind but a well trimmed boat will be much more stable up wind so the gyro would not be as critical. It could theoretically slow the boat a bit by causing excessive rudder action.