Why does a moving bicycle stay upright?

Answer: Steering keeps moving the wheels back under the bike

The rider's weight stays perfectly centered — No rider stays perfectly centered; a bike is always tipping slightly one way or the other. It's the constant small steering corrections that catch those tips.

Steering keeps moving the wheels back under the bike ✓ — A moving bike stays up because it steers into any lean, bringing the wheels back under its center of mass. The rider does this, and a well-designed bike even does it on its own, since leaning makes the front wheel turn the same way.

Spinning wheels act as gyroscopes that hold it up — Spinning wheels help only a little: experimental bikes with their gyroscopic effect canceled out still balanced themselves. Steering the wheels back under the bike matters far more.