Why do so many crystal radios have big coils?
Note how these inductors are air filled so even though they're physically large in size, that doesn't mean they also have a large inductance.
Indeed tapping off a signal is much easier with large inductors and that is I think the actual reason why they're this large.
With a smaller inductor tapping will be more difficult, the wires are not isolated, if they were, tapping off would be impossible (unless you remove the isolation first). Also the un-isolated wires cannot touch as that would short the inductor.
Also, with a smaller inductor you would get a smaller voltage (at a higher current but that doesn't help) so the output signal would be smaller resulting in a lower volume in the (high impedance, crystal) headphones.