That article, even though it was written by a physicist, sounds a lot like that apocraphal story about "scientists prove that a bumble bee can't fly! haha dumb scientists" but which is actually a story about a scientist describing how simplified models only work within the scope they were intended to apply, and fail when applied to other systems (aerodynamics models meant to apply to aircraft don't work when applied to something that flies like a bee).
Of course we know how bicycles work. People have known for a long time that gyroscopic effects don't keep bicycles upright - in fact precession would actually tend to make the bike LESS stable during steering. The study with the counter-rotating wheels just put the nail in the coffin on that one.
Tire friction matters, as mentioned, as does the geometry of a bicycle - due to the rake and trail of the fork, a bicycle in a "turning" position has a higher center of gravity than a bicycle in a "straight" position, so gravity literally pulls the bicycle straight again, by lowering it's center of gravity.
I've made and ridden a lot of "freak bikes" and you realize pretty quickly how small changes, like making a bike with no rake and trail, make it nearly unrideable, since it is no longer dynamically stable.
I suppose it may be true that no one has written up the whole model that describes the bicycle (and motorcycle's) behavior, but that doesn't mean that it is some kind of mystery, just that it hasn't all been generalized yet.