My professor debunked this in experimental physics 101. He installed counter wheels on the wheels which spun in the opposite direction (without touching the floor obviously) and was able to still ride the bike just fine.
It may play a part in the effect, but does not explain it entirely.
Well, you can debunk that debunking easily enough if you have a bike.
Stand in front of the bike and lift up the front wheel, holding one fork in each hand.
Attempt to turn the front wheel from side to side without the seat changing position at all.
Now hold the bike up with one hand and with your other hand spin the front wheel as fast as you can.
Now with the front wheel still spinning, go back to holding one fork in each hand and attempt to turn the front wheel side to side without the seat moving at all. You will find it more challenging to do and you will notice that the tendency is for the seat to move in the opposite direction from that which you turn the front wheel.
That is, you turn the front wheel towards your right side, the seat will move towards your left side.
This is the "counter steering" effect that we use in order to balance when riding a bike, and it's entirely due to gyroscopic motion.
“It is almost certain that gyro effects are important at the initial stage of steering manoeuvres. […] My point is that gyroscopic effects are not needed to keep you from falling over when you are riding in a straight line. I am not saying anything about what happens when you actively wish to steer away from straight ahead.”
It also does some calculations that show how small the gyroscopic force is compared to the weight of (bicycle plus rider)
So the gyroscopic effect isn’t necessary to balance a bike, but likely helps in making turns.
The gyroscopic effect is not necessary, and it has been demonstrated both in reality and in simulation [0].
The central principle behind why a bicycle is self balancing while in motion is the fact that it self-steers in the same direction that it is leaning, which counteracts the fall [0][1].
It may play a part in the effect, but does not explain it entirely.