Why is only the fundamental frequency component said to give useful power?
It's an overly-general statement. Obviously, with a resistive load, all frequencies transfer power.

It's really a statement about rotating machinery specifically (motors and generators). For these devices, energy at frequencies other than the fundamental is just as likely to oppose the work being done as aid it. Also, the energy of high frequencies is often wasted in the form of unwanted eddy currents, etc.