One can always test the hypothesis via an experiment:
You cannot lean without countersteering first; the lean happens because you induced a fall in the desired direction, through a tiny countersteering move. With the tiny, subconscious counter-steers, you induce only small tilts. I suspect it takes multiple small counter-steering maneuvers to "build up" a decent tilt for a sharp turn, which takes time.
Once I learned to deliberately countersteer, I then started doing it all the time. I hardly take any turn (big or small) on a bicycle without pushing forward the handle-bar on that side.
If you don't make deliberate countersteering your main steering method, so that it becomes second nature, you will not be able to count on yourself to use it in an emergency.
At low speeds on e.g. a light MTB stop pedaling, lift your butt off the saddle, lean bike far into desired side balancing your body and handlebars to keep going straight. You have most of what's needed to turn. You may countersteer but your line does not veer into the direction of countersteer, counter to illustrations.
Riding edge of a cliff is not a great idea anyway because balancing a bicycle tends to require countless small adjustments even due to pedaling alone.
It's good enough if you already understand it and know what to look for; just not the best for convincing someone who doesn't believe the physics.
A steering column which somehow turns right without your hands will induce a countersteer to the left the same way as if you had used your hands.
That's irrelevant; bicycles countersteer to keep stable, without a rider present (above a certain speed).
If the riderless experiment were repeated with a bike whose steering is locked out, it would fall down rather quick.
The bike seems to be initially leaning to the left, because it was induced into motion that way by the experimenter. Yet in spite of this, it recovers by tilting to the right.
If there was a human on that bike wanting to turn, here they would continue leaning, turning, and pedaling in the correct amounts.
But because there's no human to keep their weight leaned to the left, then 3. The left turn of the handlebars causes the road contact points to move to the left of the center of gravity, so that 4. It starts turning right.