There is no inconsistency between Maxwell's equations and Newtonian physics, per se, and Maxwell specifically ensured covariance under the Galilean transforms. Where the inconsistency actually occurs is with the constitutive laws.
In detail ...
The Maxwell equations, when written in this form
$$
= ∇×,\quad = -∇φ - \frac{∂}{∂t}\quad⇒\quad ∇· = 0,\quad ∇× + \frac{∂}{∂t} = ,\\
∇· = ρ,\quad ∇× - \frac{∂}{∂t} = \quad⇒\quad ∇· + \frac{∂ρ}{∂t} = 0,
$$
are covariant with respect to the Lorentz transforms, and the Galilean transforms ... and the 4D Euclidean transforms, because they are covariant with respect to arbitrary 4D coordinate transforms.
The constitutive laws for isotropic moving media, as written by Maxwell and later corrected by Heaviside, Thomson, etc.:
$$
= ε ( + ×),\quad = μ ( - ×)
$$
break this symmetry with the appearance of the reference velocity $$, by virtue of the specific way it enters these equations. They are only covariant under the Galilean transforms.
And, yes, they are also equivalent to what Lorentz wrote:
The "Rosetta Stone" Between Lorentz' Equations Versus The Maxwell Equations + Maxwell(-Heaviside)-Thomson Relations
Lorentz' equations were not Lorentz-covariant, but Galilean-covariant. The idea that Lorentz provided a relativistic formulation for electromagnetic theory is a widespread myth. The equations he actually wrote down (as listed in Abschnitt II of his 1895 paper, as detailed in the above link) are just Maxwell Equations + Maxwell(-Heaviside)-Thomson Relations, up to a change in notation.
To get symmetry under the (ironically named) Lorentz transforms requires modifying them to
$$
+ \frac{1}{c^2} × = ε ( + ×),\quad - \frac{1}{c^2} × = μ ( - ×),
$$
as derived (up to notation) by Einstein and Laub in 1907-1908 and independently by Minkowski in 1908 in his "Minkowski geometry" paper.
The place where the difference makes itself felt most acutely is that the Maxwell-Minkowski(-Einstein-Laub) relations are $$-independent for media where $εμ = 1/c^2$ and $|| < c$ - thereby giving you a way to explain the absence of any reference velocity in the vacuum, where $(ε,μ) = (ε_0,μ_0)$ and $ε_0μ_0 = 1/c^2$.
In contrast, the Maxwell(-Heaviside)-Thomson(-Lorentz) relations have no $$-independent version ... except for extreme media where $εμ = 0$, such as (near-)zero-permittivity media, where $ε = 0$. There might even be such a thing as (near-)zero-permeability media, where $μ = 0$. Superconductors might qualify, because of the Meissner effect.
Lorentz' formulation was also Galilean-covariant and he used $$ for $-$. Other 19th century and early 20th century literature, including Lorentz, made the distinction between the "stationary" ($ = $) and "moving" ($ ≠ $) forms of the equations formulated.
For the non-relativistic setting of Newtonian theory, the appearance of $$ is necessary, since (apart from the extreme cases cited above of $εμ = 0$ metamaterials), there is no $$-independent formulation of the constitutive laws. When Einstein titled the 1905 paper "On the electrodynamics of moving bodies", it is the discrepancy between the "stationary" and "moving" that the paper was in reference to and that was being called out. The 1905 paper showed that $$ is "superfluous" (to use Einstein's term) for media where $εμ = 1/c^2$, such as the vacuum, and the 1907-1908 follow-up papers provided the Relativistic correction to the non-relativistic version of the constitutive laws.
