Does this help to explain the fixed correspondence between steer and direction of turn? (For example, to someone who doesn't believe counter-steering is a thing.) The physics of the lean seems easier to decompose than the physics of counter-steering.
I am interested in finding a way to describe counter-steering so that it's obvious, similar to the way you described the horizontal force vs coefficient of friction.
Inertia keeps your center of mass moving straight, so your center of mass is now to the left of the wheels.
Think of it as the ground pulling the wheels out from under you.
Similarly on a bike, if you turn the wheel right, you will eventually either fall over, or turn to right (which will happen with an unconscious lean). Since the correction to lean-right is usually not done consciously, and happens very quickly most people don't realize there is an initial lean left.
Draw a picture of a leaning bike from behind. Add a horizontal line for the ground. Assume the entire bike/human system is leaning an equal amount, so put the center-of-mass somewhere in the leaning line.
If the bicycle is maintaining a constant amount of lean, then the extended line of the force applied by the ground to the bike must pass through the center of mass (i.e. in this case the angle of the force vector is the same as the angle of lean).
Now decompose the force vector into vertical and horizontal components. The vertical portion when riding on the ground (i.e. not jumping the bike), neglecting aerodynamic effects, is fixed to be the gravitational force, so it will be constant regardless of the direction of lean.
This means that the horizontal portion of the force will be equal to the the gravitational force times the tangent of the lean-angle. This also means that when the tangent of the lean-angle exceeds the coefficient of friction between the tires and pavement, you will crash.
The turn-radius can then be calculated by treating the horizontal force as the centripetal acceleration, and is indeed related to speed (faster speeds mean a larger turn radius).