Wouldn't that imply that a bicycle wheel I roll down the street by itself should fall over immediately which clearly is not the case?
Furthermore you will notice a bicycle wheel rolling down the street often won't roll in a straight line (as it starts to tilt sideways, it will turn to bring the contact patch back underneath it), so even with just the wheel, the gyroscopic stabilization is insufficient on its own to keep it upright.
Veritasium made a video about it: https://youtu.be/9cNmUNHSBac?si=8cbWXgoqg-uuVidb
At very low speeds, yes, the bike will fall over. But a bike with some minimum amount of speed can roll upright on its own just fine. You can try it yourself with your least favorite bike and an empty parking lot. All it takes is a good solid push. It has to do with bicycle frame geometry and center of mass.
Neil deGrasse Tyson explains it in this video at 2:52: https://www.youtube.com/watch?v=RIrgwWQqqts