The Potentials Phi and A are constant in time. Therefore the magnetic B and electric field E are constant in time. Therefore only the (time) Fourier component with frequency 0 exists. No photon emits.

You'll be happy to know that you, and Mills and reality all agree. In "ground" state Hydrogen, the electron does not radiate as you have deduced. However, apply the same analysis to the excited state of Hydrogen. Now, the electron orbitsphere has absorbed a photon. The electric field from this "trapped photon" partially masks the central charge, and the effective charge felt by the electron is now "fractional". Using the same equations, Mills shows that the space-time Fourier now has a sinc(x) component (i.e. sin(x)/x) which does not fall out neatly and leads to the components that are synchronous with lightspeed. Thus, the excited state is unstable and will eventually emit a photon that will return the atom to "ground" state. Once back at the ground state, the potential for those synchronous with lightspeed Fourier components disappears and the atom cannot radiate any more. Please take a look at Chap 2 in Mills' book and find the sinc(x) in equation 2.29. I think it'll will make more sense to you now.