How did you test a bike staying upright on its own?
Being stable only requires the bike to resist some reasonable force that would knock it down. It takes much more force to tip a moving bike than a stationary one.
If you have a hill without anything breakable at the bottom of it, and a bike you don't mind breaking, try pushing the bike down the hill. I don't have any hills here in Florida, but I recall that working pretty well as a kid living in more interesting terrain.
Being stable only requires the bike to resist some reasonable force that would knock it down. It takes much more force to tip a moving bike than a stationary one.