Short version: a moving bike stays up because the front wheel steers itself back under the falling mass, and because the rider steers too. Balance is a steering problem, not a spinning problem.
When the bike leans left, two things push the front wheel to steer left. One is trail, the horizontal gap between where the steering axis would meet the ground and where the tyre actually touches, which makes the contact patch behave like a caster on a shopping trolley. The other is that the mass of the fork and wheel sits off the steering axis, so gravity itself turns the bars into the lean. Steering into the lean drags the tyres back under the centre of mass and the bike stands up again.
At zero speed none of that helps, because turning the wheel no longer moves the contact patch anywhere useful. That is the entire difference.
Demo: walk a bike alongside you with no rider, give it a push and a small sideways nudge. It corrects itself and keeps rolling. Then hold the bars rigidly straight and push it again. It falls over almost immediately.