What is Wrong with Newton's Law of Gravitation?

The corner stone of classical mechanics/physics is Newton's Law of Gravitation taking the form of the Newton/Poisson Equation NPE

• $\rho =\Delta\phi$         (*)

connecting mass density $\rho (x)$ to gravitational potential $\phi (x)$, where $x$ is the space coordinate of 3d Euclidean space. 

In modern physics this mathematical model is replaced by Einstein's Equation EE in "curved space-time" of his General Theory of Relativity GR preceded by the Special Theory of Relativity SR. EE reduces to PE in flat Euclidean space without time and so EE is viewed to be a generalisation of PE into curved space-time.   

To mark a shift between classical (obsolete) physics and modern physics a lot of effort has gone into showing that NPE contradicts observations and so must be replaced by EE. In particular NPE is viewed to contradict the following consequences of SR/GR:  

1. Finite speed of light.
2. Gravitational lensing. Bending of light 
3. Time dilation. Clock rates affected by motion and gravitation.
4. LIGO detection of gravitational waves with finite speed of propagation.
5. Precession of Mercury Perihelion.  

But NPE says nothing about propagation of light or the rates of clocks and so 1-3 cannot be viewed to contradict NPE. 

As concerns 4, it is well known that a delay in the action of the gravitational pull on Earth from the Sun will make the Earth orbit away from the Sun. Finite speed of propagation of gravitational force thus appears to contradict the stable orbit of the Earth. In EE this is explained as a form of compensation of the delay in some form of prediction effectively cancelling the delay to no delay. Strange.

So the evidence against NLG shrinks down to 5 with the claim that NPE gives an incorrect prediction of the observed orbit of Mercury, while that of SR/GR is correct. But making a prediction requires input of positions and velocities of all celestial objects in the Solar system at some specific time and in addition their masses and G. To claim that NPE gives the wrong prediction in the form of a very small deviation from observation requires a very accurate NPE computation taking all celestial objects including their internal motion correctly into account. Such a computation has not been made. 

Another argument against NPE is that it requires a notion of absolute time, something which is supposed to contradicts the relative time of SR. Is this a valid argument? 

In the sequence of posts on Neo-Newtonian Gravitation I have tested the idea that the gravitational potential $\phi (x)$ is primordial from which mass density is $\rho (x)$ is delivered by the local action of the Laplacian according to (*) as if time does not enter. 

Combining NPE with Newton's 2nd Law $F=am$ law brings in a notion of time since acceleration as change of velocity per unit of time refers to time, as well as velocity as change of position per unit of time. This gives a notion of local time for each body interacting with all other bodies through the time-less gravitational potential, which does not ask for coordination of local times into a global time. Maybe Newtonian mechanics in fact does not require a notion of global time? Only a notion of a global time-less gravitational potential.