One doesn't follow from the other. We can easily transport power by making things like a chain or a fluid move back and forth, without any net travel. In a setup with a loudspeaker and a microphone as just one example the air transfers energy from one to the other without any net movement. In those cases it's clearly the movement itself which transfers the energy. Therefore energy transport by AC is no proof for the need of an electric field for energy transport.

That's not say to there is no electric field, or to deny its role in power transfer. There certainly is an electric field. But that field is intimately tied to the electrons in the conductor, and power transfer is intimately tied to movement of those electrons and the way electrons repel each other stronger when they get closer together (or other charge carriers, but in typical conductors that means electrons). You can't have one without the other.