You see, in a balanced turn, the precession will cause the rotor of the rate gyro to go to the vertical.
Hold on a minute. It may cause it to go slightly out of plane, but not vertical. The angle by which the rotor precesses will depend on the elasticity of the strings used to tie it. Use very stiff springs (with a highly geared indicator) and it will remain very close to its original axis.
The question I've always wondered about is which axis is the turn gyro spinning about (in equilibrium). Clearly it has to be perpendicular to the yaw axis of the aircraft, but it could be fore/aft or port/starboard. My (very dusty) groundschool notes indicate port/starboard, but both the construction of the cases I've seen and credible source suggest fore/aft. Anybody taken a modern turn indicator apart?