Ok, in studying of Maxwell equations we have violation of Galilean relativity. This implies necessity of other transformations which make Maxwell equations covariant (invariant in form) under this transformation. This transformation is Lorentz transformation which conserve speed of light for all observers. Question is why any other process must be Lorentz invariant, how we know that all other processes are Lorentz invariant ?
We know this from experiments.
Actually, there are quite a number of attempts and experiments to find out if really all processes are Lorentz invariant - and thus far they all came to the conclusion that (within some experimental measurement error) all processes indeed are Lorentz invariant.
An experiment to the contrary would actually break quite a number of currently accepted theoretic models.
Beyond experiment there is no other reason why a process must be Lorentz invariant. It just is so and we measure that. Of course we could also claim that nature followed certain laws and then try to deduce this from these laws, but that's not how physics is done (in mathematics this is different, however).
Update: It is quite tempting to assume that there may be logical interrelations between physical theories, which force upon nature certain properties. This thought has been mentioned in the comment section and I think it is in order to elaborate on this a bit more since it seems to be behind the question of the OP. To the discussion I can offer the following arguments:
Argument 1: In mathematics we know quite a number of different logics (intuitionistic, linear, modal, temporal, branching, non-deterministic etc). There is not just one logic. The different logics lead to different formal results. Which of these logics would you be using in your physics?
Argument 2: In physics we used to employ the traditional, dichotomic, Boolean logic for reasoning about systems. With the advent of quantum mechanics it became clear that this approach leads to numerous problems. Therefore, Birkhoff and von Neumann developed a lattice based quantum logic, where numerous laws from Boolean logic do not apply. This quantum logic is well adapted to the structures currently used in quantum physics and seems to be a better logic for describing reality. So we do have a case where even in physics a logic system was changed.
Argument 3: There are quite a number of things which do not go well with each other in physics. If you check out topics like Bell inequality, hidden variables, infinities and other (in)consistencies in quantum field theory, you will see a wide range of things, we can describe in an empirically satisfying manner, which, formally contain inconsistencies and contradictions. The art of physics is the art of pushing these problems aside so that they do not get in our way when modeling our world. Still they are there. We might not like the situation, but it is as it is.
Argument 4: Let's assume we have this wonderful theory and it is logically all consistent. All of a sudden we discover that nature does not follow our all so logical conclusions. So what? Is nature wrong? Will we punish nature for being illogical? Or would we rather humbly admit that our expectation was incorrect that nature has to behave in such a way that it looks logical to our mind?
Argument 5: Logical interrelations and deductions, of course, provide us with a string instrument in physics. We will study all logical consequences and interrelations and will test the results against experiment. This process guides our search for better theories. However, we should never assume that there is a logical reason why some physical process must behave like this or like that (or, be Lorentz invariant), just because it follows from some mathematical conclusions. So, we should happily welcome all theories and speculations, which violate energy conservation, the laws of thermodynamics, Lorentz invariance etc. However, history tells us that theories and speculations which do violate these things, most of the time quite quickly failed. So expecting Lorentz invariance from a theory in fact is a good expectation for saving time. Still, there is no fundamental reason (as in "logical")...
The path to relativity that you're describing is the historical one, but it's not the only possible one. With hindsight, it's basically a historical mistake that Einstein thought electromagnetism played some special role in the logical basis of relativity. For an example of a more modern approach, see Pal, "Nothing but relativity," https://arxiv.org/abs/physics/0302045
It's logically possible for physics to follow either Galilean relativity or Lorentz-Einstein-style relativity. However, it's not logically possible for some phenomena to follow one and some the other. Both state that (a) there is no preferred frame, and (b) the transformation between frames works in a certain way (which is different in the two cases). If one set of phenomena (say, mechanical phenomena) transformed one way and another set of phenomena (say, optical phenomena) transformed the other way, then there would be no way to keep them consistent with each other, and there would be a preferred frame. That is in fact what physicists believed before 1905, in the days of aether theories.
Even after 1905, it's actually not true that nobody ever bothered creating theories that broke Lorentz invariance. Some examples:
Hoyle's 1948 steady-state model of cosmology breaks Lorentz invariance.
The PPN framework for tests of general relativity explicitly includes Lorentz-violating terms.
In 2011, a group at CERN published a report, later found to be incorrect, of neutrinos traveling faster than $c$. This prompted a significant number of theorists to cook up Lorentz-violating theories.
Hoyle, “A New Model for the Expanding Universe,” MNRAS 108 (1948) 372, https://ui.adsabs.harvard.edu/abs/1948MNRAS.108..372H/abstract
In physics there are different theories, mathematical models, for different frameworks. Each theory starts with laws and postulates and principles, which are directly connected to experimental measurements and observations. These are extra axioms imposed on the solutions of differential equations to pick up the relevant ones, the ones that describe data, and, very important, are predictive.
All these different frameworks in their overlap regions can be shown to give correct predictions ,within the errors of the measurements. Some of the theories are emergent, as is thermodynamics emergent from statistical mechanics. Some of the theories give very good results to first order, even though based on galilean transformations, as are Newtonian mechanics. Special ( Lorenz transformations) and General relativity are higher order corrections to GPS and planetary, galactic and cluster of galaxies orbits.
So there is a hierarchy in the frameworks that mainstream physics uses: there is quantum mechanics with Lorenz transformations incorporated, using four vectors all the way, which is proposed as the underlying framework for all physics. In this sense Lorenz transformations are inherent in all the framework theories because the theories are consistent in the regions of overlap.
Why were Lorenz transformations incorporated in quantum mechanics? Because theories using them describe the nuclear and particle data exceedingly well, and the theories are continually validated, i.e. are correct in their predictions.
It has to do with absolute versus relative motion. Issac Newton is the one who put forth the concept of absolute spacetime (ether concept, also instantaneous action). People really didn't question until late. In 19th century Faraday and Ernst Mach (The Science of Mechanics) severely criticized this absoluteness of spacetime. By the time of Einstein, Maxwell's theory was available for him to bring a paradigm shift. As we now know relative spacetime concept incorporates homogeneity and isotropy concepts as well as time and space translation invariance which are nothing but Poincaré group. That means physics must be the same everywhere in the cosmos: the so called Lorentz invariance. There are no preferred directions (no absolute space) and no preferred clock (no absolute time). At least until now this is the better understanding available.