Poisson's equation $\Delta \Phi = 4 \pi G \rho$ is not Poincare invariant, hence the need for a new theory of gravity after Special Relativity (SR).
The motivation for gravity as curvature in General Relativity (GR) appears to have been the equivalence principle, which is true in Newtonian gravity also. Indeed, the idea of gravity as curvature appears also as a reformulation (or perhaps mild generalisation) of Newtonian gravity called Newton-Cartan theory.
Is there something about Lorentzian geometry and about curved geometry that makes the two play particularly well together as opposed to Newtonian geometry and curved geometry?
Is gravity as curvature the best view of any theory satisfying the equivalence principle, regardless of whether the theory has Lorentzian or Newtonian geometry?