Man on a bike - leans into a corner. How?

[rcgldr]

If the leaning back action is done slowly, the swing chain and the body take a new orientation, however, with the CG still staying vertically below the suspension point.

It's easier to visualize this with a person standing on the swing. The person leans back, pushing the swing at his/her feet forwards, and pulling the chain at his/her hands backwards. The forces exerted by the person cancel, but the tension in both segments of the chain above and below the person's hands have a forwards component of force at the persons hands. The tension below the person's hands is countered by the persons compression, but the tension above isn't countered, resulting in forwards component of force in the chain at the person's hands, and a backwards component of force in the chain at the supporting bar. If the supporting bar is fixed in place (attached to the earth), the result is a forwards movement of the person and swing (and a tiny movment of the Earth backwards).

As someone points out above the initiation and control of a turn (bicycle or motorcycle) is dependent on the speed.

and the steering geometry. The steering geometry can be designed with sufficient trail (distance from contact patch to where steering pivot axis intercepts the ground), to be stable at very slow speeds.

Things are less confusing when you get rid of the extra wheel and only have one wheel, a unicycle. The unicycle oriented web sites never get into these debates. It's accepted that leaning requires countersteering at all speeds. There are various methods of turning though, using arms and/or pedal pressure to control direction and/or turning of the wheel.

I would say there is no such thing as an isolated system.

An isolated system is an abstract concept, so it may not be "real", but the abstract concept can be used to simplify the physics of an interaction.