With an AVCS gyro, on the ground you get the behavior your talking about.

You give stick input. The gyro (not the stick) moves the servo. The helicopter doesn't move to match what the command was so the servo stays offset. If you were to rotate the helicopter to the new heading the servo would end up back to center.

In rate mode a gyro just dampens movements which is why it returns to center.

In flight the heli would move to follow the gyro command and the servo would re-center.