Bicycles are another commonly-misunderstood system covered on cam.ac.uk. A lot of people seem to think they stay upright by gyroscopic effects. It's more a case of the rider making corrections, and when there is no rider (or hands-free) it works by a feedback loop involving the geometry of the forks.
You can hold a bike wheel in your hand vertically and spin it, when you try to move it to a horizontal orientation you can feel the extent of the gyro force. It's significant.
It's enough to keep a loose rolling wheel upright by itself, but nowhere near strong enough to keep an entire bicycle plus rider upright. I don't think rim mass has much effect on handling - it's all about the geometry.
The force is proportional to speed, you can countersteer a bicycle just the same as a motorcycle. There are 2 separate steering techniques, one for high speed (countersteering) and one for low speed (steering) on motorcycles and bicycles - that results from gyroscopic wheel force. The geometry doesn't change, only thing that changes as speed increases on the bike is the gyro force.
https://www.cam.ac.uk/research/discussion/opinion-how-does-a...
Gyroscopic effect is very minor unless you're going very fast, i.e. motorbike speeds and motorbike weight wheels/tyres.