Also there have been tests where a perfectly balanced human-size/weight dummy is attached to a bike, and it falls over even when it is travelling at a speed that a human could ride easily. That suggests that minute adjustments to balance and steering are a major factor in staying upright.
That's the gyroscopic effect. If the wheel is rotating and you twist the axle side-to-side (in a horizontal plane), there is a force that tries to twist the axle up-and-down (in a vertical plane). It acts so as to correct a tendency to fall over: if you start to lean to the left and the front axle turns to the left, the gyroscopic force acts to try to move the bike back to vertical, and its magnitude varies with speed.
I can see why people though that this was the only force that mattered, since it does play a small part in keeping balance - it's just that it's not enough on it own.
I read of an experiment where a bike was fitted with wheels that had discs of equal mass to the wheel that were rotated (electrically) in the opposite direction so as to cancel out any gyroscopic force. And the bike was still rideable, though it was slightly harder to keep one's balance.