My guess is that the Furuta Pendulum is fully controllable when the input is the torque, and not the underlying DC motor characteristics.
If you replace the entire motor with a direct torque input, the system is controllable and the controllability matrix has rank 4:
At the same time if you use the simplified DC motor model current is linearly related to torque, so if you feed a current sources to the motor (instead of voltage) you get as well a fully controllable system:
I may be wrong, but I don't think you can get a proper controllability matrix if you include two cascading system (external loop: arm position, internal loop: DC motor torque) as both open loop, like you were doing. You need to examine both system independently, the position loop assuming a perfect dc motor (eg torque (or current) fully controllable)
Hope this sparks some thoughts - it has been many years since my control system lectures so take everything with a grain of salt.