The description of the mechanics and physics of bicycles is nearly completely wrong.
The largest contributing factor to the stability of bicycles is height of the CG above the ground coupled with the amount of trail that the contact point of the front wheel has behind the intersection of the steering axis with the ground.
A bicycle has two points of contact with the ground. As long as the CG (center of gravity) is directly above that line, the bicycle cannot fall over. Ir the CG is to one side of the line, there is a sideways force on the front tire where it touches the ground. The force is behind the steering axis and tends to turn the front wheel so that the line between its contact point and the rear wheel contact point is once-again under the CG.
To see the above described effect, take a bicycle and hold it upright by the saddle with the steering straight. Then lean the bicycle to one side. You'll see that the front wheel turns in the direction required for it to steer back under the CG. In the video, where the curve is drawn showing the path of the man on the bicycle, it's easily seen that he is also leaning slightly from side-to-side in coordination with the weave.
Experiments have been done with counter-rotating wheels to counteract the gyroscopic effect and the bicycles are just as rideable. Friction as a means to force the front wheel straight can be seen on shopping carts, which have enough wheels that steering and avoidance of falling over are controlled independently.
Another example is the unicycle - the rider, when holding a position, is constantly moving the wheel to be directly beneath the rider. The advantage of a bicycle over a unicycle is that falling front-to-rear is prevented by the extra wheel, but side-to-side is the same as a unicycle - keep the wheels under the CG.
The reason the Penny farthing has so large a wheel is not for stability, but for increasing the ratio of distance traveled per turn of pedal, something now done by chains and sprockets.