There's no problem with the non-invariance of D'Alembert equation for mechanical waves, if I understand what you mean, because mechanical waves do have a preferred inertial frame, an "aether".

For example, a sound wave in a fluid satisfies the wave equation with speed: $$c^2=(\frac{\partial p}{\partial \rho})_s$$ in the rest frame of the fluid.

The point is that Maxwell's equations are supposed to be valid in every inertial frame of reference. Since, in vacuum, they lead to the wave equation, the wave equation must be valid in every inertial frame, that's the problem.

There's no problem with the non-invariance of D'Alembert equation for mechanical waves, if I understand what you mean, because mechanical waves do have a preferred inertial frame, an "aether".

For example, a sound wave in a fluid satisfies the wave equation with speed: $$c^2=\left(\frac{\partial p}{\partial \rho}\right)_s$$ in the rest frame of the fluid.

The point is that Maxwell's equations are supposed to be valid in every inertial frame of reference. Since, in vacuum, they lead to the wave equation, the wave equation must be valid in every inertial frame, that's the problem.