It’s not that. Gyroscopic action exists of course, but it’s fairly weak against the weight of your body. Balancing a bicycle is just like balancing an umbrella on your finger, except you can easily move your finger any direction you need. To move the bicycle sideways, you need to already be moving forward.
it really is that though, have you not done the rotating wheel experiment? it is fucking hard to tilt the axis of a wheel rotating at a speed which is comparable to biking speed. come to think of it maybe not:
Track stands! Not a contradiction to your statement at all though: you need to be moving just ever so slightly.
With a fixie it’s easy, because you can pedal forwards and backwards in tiny amounts. With a freewheel, it’s trickier but you get the hang of it with practice. Ideally you’ll have an incline, so you pedal forward to go forward, and ease up to slide back. After some practice I can use the raised reflective paint from e.g. crosswalks as the “incline.” This miniscule motion is enough to balance — and like you said, it ain’t the angular momentum that does it.