The gyroscopic effect of wheels has a minimal impact on the tendency of things like bicycles to remain balanced; the majority of the effect comes from the rake of the front wheel (i.e. that the point where it touches the ground is ahead of the pivot point of the front post).

That gyroscopic effect of the wheel works well when the whole device is wheel (think hoop rolling), but when you start adding weight that's not spinning (like someone riding a scooter), the effect falls off.

Edit: it seems that neither the gyroscopic effect nor rake are sufficient to explain bicycles' self-stability: https://en.wikipedia.org/wiki/Two-mass-skate_bicycle