Postulates of Newton's vs Einstein's Theory of Gravitation?

Modern physicists claim that Newton's Theory of Gravitation NTG must be replaced by Einstein's General Theory of Relativity GR as a mathematical model of motion on scales from planetary to galactic. 

The basic postulate of NTG is the Poisson equation $\Delta\phi =\rho$ connecting gravitational potential $\phi (x)$ with mass density $\rho (x)$ in terms of a Euclidean space coordinate $x$. This relation can mathematically be deduced from the following two physical principles

1. Gravitational force $F(x)$ is conservative and as such is the gradient of a potential: $F(x)=-\nabla\phi (x)$. 
2. The divergence of $F(x)$ expresses presence of mass density (as a sink): $\nabla\cdot F(x)=-\rho (x)$.

Together 1+2 give $\Delta\phi =\rho$. Voila! It is very hard to argue that 1 or 2 are not valid if there is anything like gravitational force giving rise to the motion of celestial objects. If 1 is not valid, then energy can be created out of nothing, and if 2 is not valid then gravitational force can created out of nothing. 

It thus appears impossible to argue that 1 and 2 are not correct, that is, impossible to argue that NTG is not correct. In particular, in NTG as a consequence of 1+2:

• gravitational mass = inertial mass   (E) 

If there is anything in physics for which questioning lacks reason, it is NTG as a consequence of 1+2. Yet, that is what Einstein did, even if he excused himself by: Newton, forgive me! 

What are then the basic postulates of GR?  (which have to differ from 1+2 unless Einstein=Newton).

A search does not give any clear answer like 1+2 for NTG, but here is one option commonly viewed to capture (some of) the essence of GR:

• General Principle of Relativity GPR: Physical laws take the same form for all observers independent of motion, also named covariance. 
• Equivalence Principle = E.

Gravitational force has a fundamental role in NTG, but no role in GR. Nevertheless it is agreed that GR reduces to NTG except in utterly extreme situations such as a merge of two black holes. So somehow GR must include also 1 and 2, and so there must be something more than GPR+E. But what? 

Moreover: If GR includes 1+2, then E appears redundant, since it is a consequence of 1+2. 

So we are left in confusion: What are the basic postulates of GR?

Note that Einstein uses GPR to discriminate laws of physics which take different mathematical form in different coordinate systems, thus discriminating almost all laws of physics as not proper laws of physics. Einstein can thus discriminate NTG on the ground that it is not Lorentz invariant. Newton would respond by saying that NTG is Galilean invariant and that is good enough and there is no reason to ask for Lorentz invariance. What would Einstein say? Any suggestion?