Why Not GUT/TOE as Newton+Maxwell+Schrödinger?

There is a Grand Unified Theory of physics GUT (or Theory of Everything TOE) in the form of Newton's Mechanics + Maxwell's Electromagnetics + Quantum Mechanics expressed as partial differential equations over continua of 3d Euclidean space coordinate systems and time:  

1. Newton's equations of macroscopic motion of matter/mass subject to gravitation.  
2. Maxwell's equations for light without mass/charge and matter with charge.
3. Schrödinger's equations for microscopics of electromagnetics.

But this form of GUT is rejected by modern physicists following Einstein in his pursuit of replacing Newton's Mechanics by his General Theory of Relativity GR from 1915 initiated by his Special Theory of Relativity SR from 1905. 

Einstein was driven by a fixed idea (strike of genius?) that physical laws must take the same form independent of the choice of coordinate system, named covariance. He found this to be the case for Maxwell's equations taking the same mathematical form under Lorentz transformations, but not Galilean transformations. Einstein observed the opposite to be true for Newton's equations and so, seemingly out of the blue, he decided that Newton's Mechanics had to be replaced by his new theory of relativistic mechanics in the form of SR/GR. This was a monumental proclamation which was for a long time met with much skepticism, but gradually through clever marketing has come to represent the major advancement offered by modern physics as opposed to classical Newtonian physics. 

Einstein’s leading idea that mathematical expressions (formulas, equations) of physical laws must literally letter by letter look the same in different coordinate systems, is absurd. It is the same as saying that there can be only one language word by word letter by letter, describing a common physical world. Of course the choice of coordinate system in general will affect the letters of the mathematical expression of a physical law. Nevertheless Einstein used his fixed idea to discriminate Newton’s equations because they changed form under Lorentz transformations. And after 50 years of brooding modern physicists jumped the band wagon against Newton. 

Critics point to the following consequences replacing Newton by Einstein/SR/GR:

• SR/GR is claimed to differ from Newton's Mechanics only in utterly extreme situations, such as black holes or space ships approaching the speed of light, so extreme that even SR/GR falls short.
• SR/GR is incompatible with quantum mechanics, and thus makes modern physics into non-physics without a GUT.
• SR/GR is understood by very few, if any, and so its practise is sparse, while Newton's Mechanics is understood and practised by many.    
• Presented experimental evidence concern detection of extremely weak signals from extremely massive violent large scale cosmological phenomena, such as gravitational waves from the merger of two black holes in the LIGO experiment.

In this light the following questions come up: 

• Is it necessary to replace Newton by Einstein? 
• Is GUT in the form of Newton+Maxwell+Schrödinger impossible? 

A modern physicist would answer Yes! to both questions, even at the cost of giving up the idea of GUT. A modern physicist will then have to motivate why Newton must be replaced by Einstein,  even if it means a collapse of physics as science. 

Einstein is famous for his "thought experiments" as a chain of arguments within a theory such as SR, twisted to serve as external/experimental confirmation of the correctness of the theory. In particular, the constancy of the speed of light (in vacuum) is an agreement resulting from the thought experiment of agreeing to use the Lorentz transformation, which allows SR to prove itself to be true or allowing Einstein with 100% confidence to proclaim that nothing can prove SR to not be true!  Because it is an agreement! 

But agreements can be broken, and what happens if the agreement to coordinate observations in different inertial systems according to the Lorentz transformation, is given up? What compels observers to follow such a dictate?