Yes the motor should spin, its a really simple drive mechanism, the read head is fixed to read the disk at the same time it rotates, this is why its crucial that the spindle is aligned with the head. a normal floppy drive would have a way of moving the read head independent of the rotation.
Sounds to me that you have a power issue, use your meter to check the power from the batteries is getting to the PCB and also to the motor, the black and red wires that are soldered to the motor.
The drive also has 3 switches that control the drive, so you'll need a disk in and the sled to be down before it powers up.