differential geometry - Is there an accepted axiomatic approach to general relativity?

I am reading Steven Weinberg's book Gravitation and Cosmology. He makes a big deal out of the equivalence principle and showed a bunch of deductions you can make based on it. This surprised me since other books I have read haven't emphasized it as much.

My Question:

Is there an accepted set of axioms or principles that constitute the core premises of GR from which many, most or all relevant properties can be deduced?

Answer

General relativity can be constructed from the following principles:

1. 
   

   The Principle of Equivalence

   
   


2. 
   

   Vanishing torsion assumption ($\nabla_XY-\nabla_YX=[X,Y]$)

   
   


3. 
   
   

   The Poisson equation (or any other equivalent Newtonian mechanics equation)

   
   

Explanations:

1. 
   

   The Equivalence Principle can be used to show that spacetime is locally Minkowskian, i.e. the laws of special relativity hold in an infinitesimal region around a freely-falling observer. This is equivalent to the mathematical idea that a manifold of dimension $n$ is locally homeomorphic to $\mathbb{R}^n$. This allows to do two things (that I can think of at the moment). We conclude that spacetime is a manifold. We also can make the substitutions $\eta\rightarrow g$ and $\partial\rightarrow\nabla$, which yields the correct (there are exceptions) GR equations.

   
   


2. 
   

   This is required for the geodesic equation to be obtainable from a variational principle because it implies the Christoffel symbols are symmetric. This condition is relaxed in certain theories such as Einstein-Cartan theory or string theory.

   
   
   


3. 
   

   Simply put, we need this equation to fix the constants in Einstein's equation.

   
   

All treatments of GR use the Principle of Equivalence. Weinberg's treatment especially so. The reason for this has to do with his background as a physicist. Weinberg was (and is) one of the greatest particle physicists alive. His dream was to write a coherent quantum field theory for gravity. In his mind, $g_{\mu\nu}$ being called the metric tensor is an "antiquated" term left over from when Einstein learned differential geometry from his friend Grossmann and Riemann & co.'s old papers$^1$. In Weinberg's mind, $g_{\mu\nu}$ is just the graviton field, and any connection to geometry is purely formal$^2$. In texts such as Carroll, Straumann or Wald, they use the EP to make the connection $$\tag{1}\text{Equivalence Principle}\implies\text{Spacetime is a manifold}$$ From that point on, spacetime being a manifold is assumed. Weinberg, however, was of the opinion that gravity had nothing to do with geometry and manifolds and this mathematical description was a pure formality. He has to stress the EP because philosophically he didn't accept (1).

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$^1$ See the first paragraph of section 6.9.

$^2$ See, for instance, page 77, where he calls the geodesic equation a mere formal analogy to geometry.