Why does the attitude indicator gyro not precess the same way that a directional gyro does?
The self erection device in the attitude gyro that levels it when you start up on the ground also keeps it erect (its axis pointed at the center of the earth) when precession tends to send it off kilter during flight.
Air driven gyros use a pendulous vane arrangement that controls the exhaust ports for the air being sucked through, and which always work to keep the gyro aligned with vertical when stabilized in non-accelerated flight.  This system is imperfect, because the pendulous control system is influenced by accelerations and causes its own errors, but works well enough for light aircraft.