Is the model of light as an electromagnetic wave a coincidental success? I ask this question first and foremost because of quantum mechanics, in which the electromagnetic mechanism is modeled with a quantum field, not an effect of the relative motions of continuous excitations in a charge field.
Are then the predicted waves in the electric field just a coincidental success due to our assumption that there exists a persistent charge field surrounding electric objects?
And if it is do we only continue to refer to light as such because of its usefulness?
If you look up a plot of what a light wave is the first result you'll see that one of the axes is electric field strength, which as I know it is not how quantum mechanics describes it.
Well, as always the confusion is probably due to my ignorance.
So any help in alleviating it would be much appreciated!
But before any answers, I'll add that I'm well aware that most of this has to do with its convenience in the situation you're trying to model, but my question more so relates to which is a more accurate description of reality.
 A: Electromagnetism as described by quantum field theory is referred to as quantum electrodynamics (QED).  You can actually show through a careful derivation of the QED Lagrangian--accomplished by imposing a U(1) gauge symmetry onto the Dirac Lagrangian--that the traditional electrodynamics governed by Maxwell's equations is consistent with QED. In fact, by choosing the correct gauge, we can directly derive Maxwell's equations from the equations of motion generated by the QED Lagrangian.
This is due to the fact that in order to make the QED Lagrangian gauge invariant we introduce the covariant derivative: $$D_{\mu} = \partial_{\mu} -\frac{i}{\hbar} q A_{\mu},$$ which introduces the gauge vector field $A_{\mu}$ into our Lagrangian. This turns out to correspond exactly to the electromagnetic 4-potential with the correct choice of gauge: $A_{\mu} = (V/c, \vec{A})$.
The final form of the QED Lagrangian contains the familiar electromagnetic tensor from electrodynamics $F^{\mu \nu} = \partial_{\mu} A_{\nu} - \partial_{\nu} A_{\mu}$:
$$L = \bar{\psi}(i \hbar c \gamma^{\mu} D_{\mu} - m c^2)\psi - \frac{1}{4} F_{\mu \nu}F^{\mu\nu}.$$
One of the resulting Lagrangian equations of motion is $$\partial_{\nu} F^{\mu \nu} = -e c \bar{\psi} \gamma^{\mu}\psi = -e j^{\mu}$$ for the 4-current $j^{\mu}$.
This is the tensor formulation of Maxwell's equations from which you can write down the traditional four Maxwell's equations and then derive all electrodynamics and the wave depiction of light.
A: Wave equation and EM field are central to theory of light in both classical and quantum theory. Maxwell's equations still hold in quantum field theory.
Theory of light just becomes mathematically more complicated and conceptually problematic in quantum field theory, due to fields becoming infinite dimensional operators instead of vectors, that do not commute in general, and due to appearance of infinities and various schemes needed to avoid them in results. Equations of motion of field in QT still allow for wave solutions. Quantum EM field retains many properties of classical EM field, such as the same speed of propagation, interference, but it also has additional features such as quantized states and the theory provides only probabilistic predictions about future. You can view quantum theory of light as extension of classical theory of light that has more complicated calculations and allows us to calculate things that we don't know how to calculate in classical theory, such as the Lamb shift or Compton's scattering.
A: 
I ask this question first and foremost because of quantum mechanics, in which the electromagnetic mechanism is modeled with a quantum field, not an effect of the relative motions of continuous excitations in a charge field.

Your confusion comes because the concept of "field" has a different use in classical physics than in the quantum realm. Quantization evolved slowly mathematically, starting with what is sometimes called first quantization, solving  the Schrodinger wave equation with appropriate potentials gave the bound states of the atoms that were studied with their spectra and the corresponding series.
Second quantization introduces the concept of fields in the quantum realm.

Second quantization, also referred to as occupation number representation, is a formalism used to describe and analyze quantum many-body systems. In quantum field theory, it is known as canonical quantization, in which the fields (typically as the wave functions of matter) are thought of as field operators, in a manner similar to how the physical quantities (position, momentum, etc.) are thought of as operators in first quantization.

Italics mine.
So the word "field" is defined  in a  different manner in quantum physics than in classical .
Your question:

Are then the predicted waves in the electric field just a coincidental success due to our assumption that there exists a persistent charge field surrounding electric objects?

The answer is no . The success is due because in the case of the electromagnetic interactions the same equations, Maxwell equations,  describe  the classical (electromagnetic wave) and the quantum level with its wave functions and the probabilities derived from them.
To describe quantum mechanically the  photon  a quantized version of the Maxwell equation is used, that is why the $A_μ$  appears in the answer by klippo  . See for example this version.
