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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?

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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?

Yes, the key facet of Lorentzian geometry that makes it good for geometrizing gravity is that it has a single metric for spacetime. As you mentioned, since it respects the equivalence principle, Newtonian gravity can also be geometrized as was shown in Newton-Cartan theory. However, the fit is not as natural and the chief symptom is that there is no longer a single spacetime metric, but rather a pair of degenerate metrics. In general relativity spacelike intervals and timelike intervals are different, but unified and treated on the same footing as particular values of the same metric. Not so in Newton-Cartan gravity where they are not unified at all.

Is gravity as curvature the best view of any theory satisfying the equivalence principle, regardless of whether the theory has Lorentzian or Newtonian geometry?

“Best” is a judgement call, typically based on aesthetic or philosophical preferences. I don’t have a strong opinion. If pressured to provide an opinion I would generally prefer Newtonian gravity simply for practical ease of computation. What I can say is that as far as I know any theory of gravity which satisfies the equivalence principle can be geometrized. There may be perfectly valid reasons that doing so is not the “best” choice for a particular purpose.

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