Geoff,
    Your understanding is indeed correct.  Only for drive frequencies
above the natural frequency of the instrument does the mass not move
relative to the inertial frame (instrument behaving as a vibrometer).
At frequencies below the natural frequency, it is the motion of the mass
relative to the inertial frame that limits sensitivity.  By making the
natural frequency lower, the amplitude of this mass motion decreases,
which increases the sensitivity (and is the reason for the low frequency
sensitivity to displacement being proportional to the square of the
natural period).  At very low drive frequency the pendulum aligns itself
with the vector sum of the earth's field vector g and the negative of
the drive acceleration vector a; i.e., the angle of deflection in
radians is given by -a/g (for a small compared to g).  If there were no
mass motion at low drive frequency, the pendulum would have no
acceleration response whatsoever.  Instead of the small deflection angle
in radians being -a/g, zero inertial mass motion would require the angle
to be identically zero.
    One can demonstrate the difference between high frequency stationary
mass and low frequency moving mass very easily with a simple pendulum (a
nut and a string).  Hold the pendulum with your hand at the top of the
string supporting the nut on the other end.  Move your hand rapidly back
and forth and you will see that the bob doesn't move--being at rest with
respect to the room but with a motion relative to your hand that is out
of phase by 180 degrees.  Now move the string back and forth slowly and
you will see the bob is in phase (in a relative sense) with your hand's
motion (and it most certainly is not at rest relative to the room).
     Randall