Does the reduced copper utilisation affect efficiency much when
considering other factors like output transformer losses?
Yes. With a center-tapped coil only one half is utilized at a time. To get the same number of turns in half the space the wire must be thinner so its resistance is higher, requiring more voltage for the same current and cone movement. In most speakers the voice coil resistance is not much lower than the nominal impedance. That resistance wastes power as heat, so a center-tapped speaker would probably be about half as efficient as a normal one (so perhaps 0.5% instead of 1%).
The resistance of an audio transformer winding is usually only a small fraction of the load impedance, so the reduction in efficiency due to having a center-tapped primary on the output transformer should be much less than using a center-tapped speaker.
Is a centre tapped voice coil difficult to manufacture?
It certainly would be more difficult than a single coil. An extra terminal, extra wire going to the cone, and a way to join it to the center tap. Winding and assembly would be more difficult and reliability could suffer. Having to manufacture special center-tapped versions of each speaker could reduce economy of scale and complicate inventories.
Another disadvantage of a center-tapped speaker is having to run 3 wires to it. This is a problem if you want external speakers (50% more wire, 3 terminals instead of 2, possibility of wrong connection) or a headphone jack (would need special 'center-tapped' headphones, or a separate headphone amplifier). All-in-all a lot of complication just to eliminate one or two transformers.
Why is this not done today?
With modern ICs, producing push-pull output without transformers is no problem. 'Quasi-complimentary' designs that do it with NPN output transistors have been made for many years. A class D bridge amp can produce more than 4 times the sound output on the same voltage with much higher efficiency, takes up less board space and doesn't need a big heat sink - or a center-tapped speaker.