Why does the Feynman propagator for a photon model correctly Coulomb's law?

https://physics.stackexchange.com/questions/44418/are-the-maxwells-equations-enough-to-derive-the-law-of-coulomb

The Feynman propagator for a photon is derived from the Klein-Gordon equation.

The Klein-Gordon equation is analogous to the wave equation (for the electric field E) which one can derive from Maxwell's equations.

https://en.wikipedia.org/wiki/Electromagnetic_wave_equation

One obtains the Feynman propagator from the following question: 

What is the "response" of a wave equation to a Dirac delta like source impulse at a point in space at a point in time?

https://en.wikipedia.org/wiki/Green%27s_function

The impulse response is called the Green's function.

https://physics.stackexchange.com/questions/279723/how-to-obtain-the-explicit-form-of-greens-function-of-the-klein-gordon-equation

We know that Feynman diagrams correctly model the Coulomb scattering of electrons and positrons. The scattering in classically governed by the Coulomb force. Why do Feynman diagrams work? They are derived from a wave equation, not from the Coulomb force equation.