I have never had one of these pumps apart, but there are probably bearings or some type of sleeve that the inside of the motor rotates. If these surfaces are corroded or rusted, there will be static friction to keep it from moving. Once it breaks loose, the friction is much less and it continues to work.
That is one thought.
Another, there are probably some sort of brushes inside (it is a dc motor), and the brushes are dirty or corroded and not conducting electricity, thus, the motor (pump) does not rotate. Once broken loose, the brushes make better contact.
If you have the pump, cut it apart and check it out. The answer may be very apparent.
Keep us posted.