We have a series of papers on exactly the topic of this discussion, but a bit more narrowly defined. That is, how and why can a bicycle balance itself?
In short, how does a moving bicycle balance itself? For a variety of complicated reasons it steers in the same direction as it falls. And, if you will excuse the sloppy informal physics language, because of the resulting curved path, thecentrifugal forces, push it back upright. What complicated reasons? Partially from the trail (or castor effects), partially from the angular momentum of the spinning wheels, and partially from other effects related to geometry and mass distribution. But there is no simple single necessary mechanism that we know of. For example, our paper in Science Magazine shows that a bicycle can be self-stable without noany castor (no trail) and with no spin angular momentum of the front wheels.
We have written several papers and supporting documents. And we have in these a pretty exhaustive coverage of the literature. So if you want to know what we think, what others have thought, and what we think about what they thought, it's all there. I don't think you will know of some important reference that we have not reviewed and described. You can start with my wwwweb page http://ruina.tam.cornell.edu (or google ruina bicycle or google schwabb bicycle.
The www sitewebsite includes photos and videos including simple explanations of some of these things.