electromagnetism - Galilean Relativity and Electrodynamics

Consider the following:

On the one hand, the principle of relativity, by Galileo, (totally applied to the Newtonian mechanics) says:

There is no mechanical experiment that you could perform to measure the difference between inertial frames of reference.

On the other hand, Maxwell's equations (or the laws of electrodynamics; laws of motion) under the principle of relativity, by Galileo, yield a non-equivalence of inertial frames of reference.

My question is: From Maxwell's equations we get an electromagnetic wave. By asking which frame of reference the wave has the velocity of $c$ we then realize that the aether is this reference. We all know that this is wrong today, but, from the perspective of a physicist of XIX century, when we try to measure the velocity of $c$ in a moving frame (with respect to the aether frame) with the Galileo's relativity principle, we then realize that the speed $c$ is different, say: $\displaystyle c' = v_{s} + c$ (*)

Is this formula $\displaystyle c' = v_{s} + c\,$ another way to verify that Maxwell's equations are different in different frames? (**)

(*) where $c$ is the speed of light with respect to the rest frame of reference with respect to the aether, $v_{s}$ is a velocity of a moving frame $S$ with respect to the aether and $c'$ is the speed of light with respect to the S reference frame.

(**) and then there is no equivalence of inertial frames for electromagnetism; and then the physics is different in one reference at rest with respect to the aether compared to a moving one, also with respect to the ether; and then there is one "absolute" frame where Maxwell's equations hold: the aether frame; and then there exists a type of an experiment that could mesure the absolute motion; and then this contradicts the principle of relativity for electromagnetism.

Answer

First of all, the Galilean and Newtonian spacetimes are not completely the same. The Galilean spacetime is a tuple $(\mathbb{R}^4,t_{ab},h^{ab},\nabla)$ (see Galilean spacetime interval?) while the Newtonian spacetime is a tuple $(\mathbb{R}^4,t_{ab},h^{ab},\nabla,\lambda^a)$ where $\lambda^a$ is a field that adds the preferred frame of rest:

$$\lambda^a=\left(\dfrac{\partial}{\partial t}\right)^a$$

In other words, the Galilean approach is relativity, which (with the Galilean transformation) is incompatible with the Maxwell's equations, while Newton adds the rest frame, which can be viewed as the frame of the aether. This approach allows adding electromagnetism to mechanics by breaking the relativity principle for electromagnetism.

Your conclusions that the Maxwell equations are not invariant under the Galilean trabsformation is correct and well known. The idea of the preferred rest frame of the aether was dismissed by the Michelson–Morley experiment that established the independence of the speed of light from the frame of reference. Lorentz has shown that changing the coordinate transformation from Galilean to that, which Poincaré later named after Lorentz, would introduce the time dilation and length contraction to the aether and make the speed of light independent of the frame. In other words, the aether became undetectable. In turn Einstein has pointed out that the Lorentz transformation can be viewed in respect to spacetime itself and the existence of the undetectable aether is no longer required. At last mechanics and electromagnetism were united under the same relativity principle.