The objective intrinsic Einstein geometrodynamical field is the induced local gauge potential-connection of the 4D translation group T4.
The induced gravity gauge potential connection here is precisely the set of four Cartan tetrad 1-forms e^I that describe the geodesic paths of Local Inertial Frames (LIFs) whose detectors are in zero g-force free-float in the spacetime as curved by matter sources.
The tetrads have the EEP automatically built in.
You CAN think of the tetrads e^I as simply another set of fields on Minkowski space-time. However, when you go to the metric tensor guv --> Christoffel Levi-Civita level of description it's Riemannian geometry.
This is a matter of taste.
The Levi-Civita connection is a complicated expression in terms of the tetrads.
i.e. discussion leading to the spin-connection eq. 2.89 in Rovelli's Quantum Gravity - a free version is online.
This is precisely analogous to the EM 4-potential Cartan 1-form A induced by localizing the U1 internal group.
You can generalize this to the de Sitter conformal group to get additional gravitational fields beyond Einstein's 1916 GR.
Didn't you prove my point with the statement?
"One does have a choice as to whether gravity is modeled geometrically as spacetime curvature, or modeled physically as a non-metrical tensor field in a flat spacetime,.." - Zielinski?
There is an arbitrary choice here (geometry) that can be separated from the physical field. That's exactly my point, with the additional reasonable suggestion to not interpret the geometry to have physical content.