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Coils are still required for axial flux motors, they're just in a different orientation [1].

Negative, mine do not. I use an air core stator where my “coils” are etched into the PCB. I don’t have to wind anything.

Infinitum?

Similar design I suppose. Air-gapped stators with printed coils on separate layers for each phase.

A coil is a shape, not a method of making the shape. But of course, there are alternatives to hand winding.