Question

why can we balance a cycle while moving but not while standing?

Answer 1

  When a bicycle is standing it is in an unstable equilibrium. A little shift (leaning) towards left side, makes the bicycle fall on the left side. This is due to the center of gravity falling outside the vertical plane containing points the wheels touch the ground.

  When a bicycle is in motion, the wheels have a rotational energy and angular momentum. This angular momentum is important. When the bicycle leans towards left side, then the center of gravity shifts towards left side. It is outside the vertical plane of the wheels' touch points on the ground . 

   Then the ground exerts a reaction force. That causes a torque on the bicycle as it is in motion. The angular momentum is altered.  Also the front wheel turns leftwards. That makes the bicycle move forward at angle to the left side. That restores the balance back. Because then the COM falls right above the touch points on the ground of the wheels.

    The automatic balance of a running bicycle is due to inertia of motion and the angular momentum of the bicycle.  The design of the bicycle is such that the front wheel is free to rotate either way. That helps the balance.