Phil Holman writes:

The greater mass of the wheel, handlebars etc. forward of the
steering axis compared to the rear of the steering axis is the
reason for the low speed steering when wheeling and leaning a
bicycle. This is the case when the bicycle is stationary also. Gyro
forces do not become dominant until much greater speeds.

I can see you are hypothesizing. Take the wheel out,spin it in your
hands, and try leaning it from side to side. If you don't believe the
strong steering forces you experience, try steering the bicycle while
holding it by the saddle when it isn't rolling. You'll notice it does
not steer.

Jobst Brandt