Why don't the wheels fall to both sides when rolling?

No, gyro principle. The angular velocity of rotation has a component in the direction of gravity. Besides the principle of gyroscope, it corrects the angle of rotating shaft. Centrifugal force is also at work.

For example, if you lean to the left, it will be like turning left. The centrifugal force must be to the right.

The center shaft+the rotating force of the wheel+the mass of the wheel itself. When the rotating force is 1/2 of the wheel, the wheel will not fall off.