>> I know even less than I thought I did

You do, and it's true for just about everyone else, certainly for me.

When I looked into it years ago I was shocked to find out how little is known about the mechanics of cycling.

There is a Brit (of course!) who has made it his life's work -- or at least the work of his spare time -- to prove wrong all the things people know™ about cycling.
I read this so long ago I don't have a link, but he built all kind of backward forks, counterrotating discs next to the front wheel to cancel out gyroscopic forces... and then he happily rode them all.

As far as straight-up cycling goes, keep in mind there are these things called ski bobs (https://en.wikipedia.org/wiki/Skibobbing) that are quite rideable, and they have no gyro forces.

Riding no-hands probably requires these forces, but I don't even know that.
Maybe there's a way to construct a frame with fork angles etc. that will steer a ski bob back on center by shifting your weight.
As a student, I was always impressed how quickly I was in the weeds in classical mechanics when the spinning top was studied. It's embarrassing.

cheers -mathias