Correct. Gyroscopic forces help a bit but are by no means essential in balancing a bicycle. This has been demonstrated experimentally by mounting additional wheels next to the regular ones on a bike but adjusted so they don't quite touch the ground and are rotating in the opposite direction at the same speed. This exactly cancels the gyroscopic effects, but the bike is still easy to ride and balance.

The basic means of balancing a bike is the same one we use when we walk. We constantly adjust our body so that our center of gravity is located above our contact point with the ground. If you feel yourself falling forward, then on the next step you put your foot farther forward to maintain your contact point under your center of gravity. Similarly, if you feel yourself falling to the left you move your feet that way on your next step. Same when riding - if you are falling a bit to the left you steer the bike in that direction so the center of gravity is again over the line joining the contact patches of the wheels and balance is restored. The bike's geometry automatically helps as well. Lean a bike a bit to the left and the front wheel will tend to turn in that direction. So when the bike starts falling there is a tendency for it to immediately steer in the direction of the fall and restore balance. This is more effective at higher speeds since then even a tiny turn results in a significant move in that direction whereas at slow speeds the bars would have to turn much more to restore the balance.