But even if there is no pressure rise across a fan, the fan does cause the flow to happen by transferring energy (or power) to the fluid.

The fan may be inducing turbulence in the fluid, and that is, indeed, adding energy to the flow.

However, in that context, the authors clearly mean that the useful action of the fan is to induce flow -- if there's no pressure rise across the fan, then for the purposes of inducing flow it may as well not be there -- so it's consuming a finite amount of power to turn, and generating zero effect. Zero effect for finite effort means zero efficiency.

But even if there is no pressure rise across a fan, the fan does cause the flow to happen by transferring energy (or power) to the fluid.

Don't assume that the flow in that graph is driven by the fan -- if the fan efficiency is going to zero, that means it is not exerting any net axial force on the airflow. That, in turn, means that the air is being driven by some other mechanism.

The fan may be inducing turbulence in the fluid, and that is, indeed, adding energy to the flow.

However, in that context, the authors clearly mean that the useful action of the fan is to induce flow -- if there's no pressure rise across the fan, then for the purposes of inducing flow it may as well not be there -- so it's consuming a finite amount of power to turn, and generating zero effect. Zero effect for finite effort means zero efficiency.