Re: Want to try to reprogram it so it feels and drives like an F1?
I think the "bicycle wheels as gyroscopes" theory is mistaken
Yep. There's been quite a lot of research into this, including various physicists both doing the calculations and creating modified bicycles which have minimal gyroscopic effects.
It's not the cyclist either; bicycles will tend to remain standing while they're moving even if they're unoccupied. This has also been demonstrated in various ways, and you can do a simple home test by standing next to a bike and giving it a good shove forward using the handlebars. If you push it straight, it should stay upright - though it may turn - for a decent distance.
It seems that bicycles are stable while moving because as they start to lean to one side or the other (i.e. as their vertical symmetry breaks) they naturally steer into the lean and turn. (This is due to the fact that bicycles are designed so the front wheel contacts the ground behind the steering axis.) That converts the lean into centripetal acceleration rather than linear.