From riding bicycles, especially recumbents I know that the reason you countersteer has mostly to do with getting your bike to lean by moving the contact patch out from under the center of gravity. With a two wheel vehicle you lean into the turn. When you turn the handlebar the tires start to move in one direction but the COG stays moving in a straight line, so you start to fall. Once the bike is leaning you can complete the turn.
Same reason you accelerate out of a turn. In a turn you tires are following a path describing a circle with gravity and turning forces balancing your COG inside but above your tires. When you accelerate in a turn the turning forces increase, forcing your COG harder to the outside of the turn, bringing your bike back upright.
Sure gyroscopic procession makes the bike a little more stable but it is a minor player in the process.
Craig