I am not a pediatrician (but my wife is). I am not a physicist (but I was a chemist). On the other hand, I am a cycler (>3000 miles [4800 km] in 1997). I do agree that it easier to fall off a bicycle at rest than one in motion. However, I believe Dr Williams' hypothesis1 about which law of physics may explain this situation is unfortunately incorrect. Although both wheels do spin when the bicycle is in motion, the amount of gyroscope force generated is incredibly small because first, the mass of the wheel's periphery (rim and tire) is relatively small and second, the number of revolutions per second is relatively few. To confirm this statement for yourself, take a bicycle, elevate the front wheel off the floor, and test how much force is required to turn the front wheel when the wheel is at rest and when the wheel is rotating as fast as you can make it go by hand. You will find precious little (if any) difference in resistance to turning the front wheel.
Gerald J. Elfenbein. The Physics of Bicycle Falls Revisited. Arch Pediatr Adolesc Med. 1998;152(4):411–412. doi: