Why isn't Whittaker's two scalar electrodynamics used when it is simpler than ordinary electrodynamics? Edmund T. Whittaker published an electrodynamics theory in 1904

Proceedings of the London Mathematical Society, Vol. 1, 1904, p. 367-372.
ON AN EXPRESSION OF THE ELECTROMAGNETIC FIELD
DUE TO ELECTRONS BY MEANS OF TWO SCALAR
POTENTIAL FUNCTIONS
By E. T. Whittaker.

which uses only two scalars potentials F and G to derive the electric and magnetic fields, in the form of dielectric displacement $\vec{d}$ and "magnetic force" $\vec{h}$ as
$$
d_x = \frac{\partial^2F}{\partial x \partial z} + \frac{1}{c}\frac{\partial^2G}{\partial y \partial t} \\ 
d_y = \frac{\partial^2F}{\partial y \partial z} - \frac{1}{c}\frac{\partial^2G}{\partial x \partial t} \\ 
d_z = \frac{\partial^2F}{\partial z^2} - \frac{1}{c^2}\frac{\partial^2G}{\partial t^2} \\ 
h_x = \frac{1}{c}\frac{\partial^2F}{\partial y\partial t} - \frac{\partial^2G}{\partial x\partial z} \\
h_y = -\frac{1}{c}\frac{\partial^2F}{\partial x\partial t} - \frac{\partial^2G}{\partial y\partial z} \\
h_z = \frac{\partial^2G}{\partial x^2} + \frac{\partial^2G}{\partial y^2} \\
$$
F and G are determined by summing over all electrons. Symbolism 116 years ago were different from today, where primed coordinates x'(t), y'(t), z'(t) denote position of the electron at time t, and $\bar{x}$ is retarded position $= x(t-r/c)$ etc.
$$
F(x, y, z, t) = \sum \frac{e}{4\pi}sinh^{-1}\frac{\bar{z}' - z}{((\bar{x}'-x)^2 + (\bar{y}' -y)^2)^½} \\
G(x, y, z, t) = \sum \frac{e}{4\pi}tan^{-1}\frac{\bar{y}' - y}{\bar{x}'-x} \\
$$
In a continuous form the two scalar potentials F and G can be
$$
F = \int_0^\pi \int_0^{2\pi}f(x \space sin(u) cos(v) + y \space sin(u)sin(v) + z \space cos(u) + ct, u, v) du dv \\
G = \int_0^\pi \int_0^{2\pi}g(x \space sin(u) cos(v) + y \space sin(u)sin(v) + z \space cos(u) + ct, u, v) du dv \\
$$
where f and g are arbitrary functions of their arguments.
 A: Classical electrodynamics is based on electrical current which is much more simple to measure than electrical charge and speed or retarded position of electrical charge separately which is used in Whittaker electrodynamics. The use of electrical current is a kind of degeneracy of two degrees of freedom into one. This risk was not given priority by the founders of classical electromagnetism.
A: Whittaker discusses this subject in the revised enlarged 1951 edition by Thomas Nelson & Son of his book to which the "archive" link disappeared. Here is a long quote from its Vol I, Chapter XIII CLASSICAL THEORY IN THE AGE OF LORENTZ, pages 409-410:

Any electromagnetic field is thus expressed in terms of the four
functions $\phi, a_x, a_y, a_z,$ (the scalar-potential and the three components of the vector-potential), and these are given by the above formulae in terms of the positions and velocities of the electrons which generate the field. It was however shown in 1904 by E. T. Whittaker, Proc. Lond. Math. Soc. 121, i (1904), p.367, that only two functions are actually necessary (in place of the four),
namely, functions F and G defined by the equations

$$F(x,y,z,t)= \frac{1}{2}\sum_e \mathrm{log} \frac{\bar r +\bar z' - z}{\bar r -(\bar z'-z)}\\
G(x,y,z,t)= -\mathfrak i \frac{1}{2}\sum_e \mathrm{log} \frac{\bar x' -x+ \mathfrak i(\bar y' - y)}{\bar x'-x -\mathfrak i(\bar y'-y)}
$$

where the summation is taken over all the electrons in the field, and
where $x'(t), y'(t), z'(t)$ denote the position of the electron at the
instant t, and $\bar x'(t)$ signifies the value of $x'$ at an instant such that a light-signal sent from the electron at this instant reaches the point $(x, y, z)$ at the instant $t$, so that $\bar x' = x'(t- \bar r/c)$. Thus $\bar x', \bar  y',\bar z'$ are known functions of $(x, y, z, t)$ when the motions of the electrons are known, and
$$ \bar r^2 = (\bar x'-x)^2+(\bar y'-y)^2+(\bar z'-z)^2$$


The electric vector $(d_x, d_y, d_z)$ and the magnetic vector $(h_x, h_y, h_z)$ are then given by the formulae

$$\begin{matrix}
d_x=\frac{\partial ^2F}{\partial x \partial z}+\frac{1}{c}\frac{\partial ^2 G}{\partial y \partial t}, & h_x=\frac{1}{c}\frac{\partial ^2 F}{\partial y \partial t}-\frac{\partial ^2 G}{\partial x \partial z}\\ 
d_y=\frac{\partial ^2F}{\partial y \partial z}-\frac{1}{c}\frac{\partial ^2 G}{\partial x \partial t}, & h_y=-\frac{1}{c}\frac{\partial ^2 F}{\partial x \partial t}-\frac{\partial ^2 G}{\partial y \partial z}\\ 
d_z=\frac{\partial ^2F}{\partial^2 z}-\frac{1}{c^2}\frac{\partial ^2 G}{\partial^2 t}, & h_z=\frac{\partial ^2 G}{\partial^2 x}+\frac{\partial ^2 G}{\partial^2 y} 
\end{matrix}$$

It will be noted that F and G are defined in terms of the positions of the electrons· alone, and do not explicitly involve their velocities.
Since in the above formulae for $\mathbf d$ and $\mathbf h$ an interchange of electric and magnetic quantities corresponds to a change of G into F and of F into G, it is clear that the two functions F and G exhibit the duality which is characteristic of electromagnetic theory : thus an electrostatic field can be described by F alone, and a magnetostatic field by G alone ; again, if the field consists of a plane wave of light, then the functions F and G correspond respectively to two plane-polarised components into which it can be resolved. Since there are an infinite number of ways of resolving a plane wave of light into two plane-polarised components, it is natural to expect that, corresponding to any given electromagnetic field, there should be an infinite number of pairs of functions F and G capable of describing it, their difference from each other depending on the choice of the axes of co-ordinates-as is in fact the case. Thus there is a physical reason why any particular pair of functions F and G should be specially related to one co-ordinate, and cannot be described by formulae symmetrically related to the three co-ordinates $(x, y, z)$.

