Einstein's Straw Man Argument to Cancel Newton's Mechanics

To clear the way for his 1905 special theory of relativity forming the revolution of modern physics, Einstein had to show that the classical physics of Newton/Leibniz, which had served mankind so amazingly well for more than 200 years, was in dire need of a revision. Einstein picked up a common idea (not carried by Leibniz) that Newton's mechanics required absolute space and absolute time and then in a thought experiment showed that absolute time in the sense of absolute simultaneity of events widely separated in space, was impossible to guarantee. Einstein argued that this could serve as a reason to cancel Newton's mechanics and replace it with relativistic mechanics coming with new notion time not asking for global simultaneity. 

But is Einstein's argument a valid argument, or is it just a straw man argument attaching a quality to Newton's mechanics which is not a necessary attribute? 

In recent posts I argue that global simultaneity is not needed for the World to go around according to Newton's mechanics. It is sufficient with a concept of local simultaneity/coexistence. In particular, Newton's theory of gravitation does not require instant action at distance formally asking for global simultaneity. 

One way to see this is to realise that the World can be made go around by explicit time-stepping, where the state $u(x,t+dt)$ of a system at a position $x$ at a next time instant $t+dt$ is computed/determined by the state at previous time $t$ with $dt$ a small time step, at positions close to $x$. The update can have the form 

•    $u(x,t+dt) = u(x,t) - dt*u(x-dx,t)*\frac{u(x,t)-u(x-dx,t)}{dx}$       (*)

where $dx > dt$ is a space step, which computes the solution of Burgers' equation modeling wave propagation in a compressible gas:

• $\frac{\partial u}{\partial t}+u*\frac{\partial u}{\partial x} =0$.  

You can follow the development of a shock propagating to the right in a shock tube by running this one-line code executing (*) by updating $u$ at $(x,t+dt)$ from values at $(x,t)$ and the nearby $(x\pm dx,t)$.

You see that only local simultaneity is needed and that Einstein's starting point that global simultaneity is needed, is a straw man argument. The consequences are far-reaching as concerns the role of Einstein's theories of relativity in modern physics.