A small amount of the flux does pass through the copper according to the saturation of the core, the width of the slot opening, and the permiability of the core.
99% of the flux lines go through the core, not the conductor.
At any given moment the bulk of the lines go through the core. But when you move a magnet from one pole to the next the lines don't break at one pole and re-form at the next. They move through the gap - and thus through the copper conductors.
Lines of force don't break. They are loops that come into being at the atomic scale and expand, or contract to a point and disappear.
The closest they come to breaking is when they "reconnect", where fields going different ways in a plasma bump into each other and reconfigure into a lower energy configuration, dumping the energy difference from their relaxation into the plasma. That is a very energetic process - the origin of solar flares, where they kick the plasma out to interplanetary distances at an appreciable fraction of the speed of light. But even then, at no point does a field line disappear in one place and reappear in another. They are ALWAYS continuous.
Answer me this: why do superconducting coils work in a transformer or motor if no flux can pass through the conductors
Because flux CAN pass through superconductors - in two different ways (both of which involve losing superconductivity in one way or another).
In type I superconductors, a strong enough magnetic field makes the superconductor stop superconducting. Then the field passes through the resistive region just fine. But the overall wire (or other conducting structure) starts exhibiting resistance again. (There are superconducting switches built on this principle. They have been used both to close superconducting loops (after they've been pumped up to carry a ring current to create a magnetic field) and to make logic gates.
In type II superconductors there's also an easier way: Flux impinging on the edge of the conductor creates a small resistive region and an eddy current resisting its further encroachment, just like with type I. But in a type II superconductor the current is free to bend. So (rather than just keeping the field out of more than a microscopic surface region) the eddy current curves and squeezes a bundle of flux down into a tight core. The flux in the tight core is enough to make THAT part of the superconductor go resistive (though the average field before the squeeze-down is far too weak to do it), but the eddy current is in the still-superconducting part and doesn't decay. The field penetrates into the superconductor and the eddy current closes into a loop around it, much like an amoeba surrounding a bit of food. If the size of the loop is smaller than the width of the conductor (and it's microscopic - a quantum phenomenon - so it generally is), and the number of them is small enough that they don't fill the conductor from side to side, the flux bottled up in the eddy current tubes can cross the conductor just fine (carrying the tubes with them to the opposite edge, where they open up and eject the field), while the overall conductor still carries current (around the little whirlpools) with zero resistance.
Type II superconductors are useful because they can carry AC with zero resistance.
Again, because magnetic fields come into being as loops expanding from a point, when you go from a superconducting ring with no current to one with a current and a resulting (extra) magnetic field through its center, field lines MUST cross the superconductor.