Furthermore, even if you could do it, why bother? "Who ordered that?" It's a waste of time. It's not an interesting question. The ball is in your court to prove me wrong here, but I think you are wasting your time.

"It reduces the gravitational field of 1916 GR to an ordinary physical field that has a completely objective definition that can be described entirely in terms of generally covariant quantities, has no intrinsic dependence on any observer's world line, and is thus not fundamentally different, physically speaking, from the electromagnetic field." - Zielnski

You are reinventing the wheel. If it ain't broke don't fix it. You keep garbling different meanings of "gravitational field" as well as garbling Newton's and Einstein's different paradigms.

1861 Maxwell Electromagnetism is a local internal symmetry U1(x) gauge field theory for electrically charged matter field actions.

F = dA

A = compensating U(1) 1-form connection field.

A = Audx^u

It is not a U(1) tensor, A's U(1) transformation has a inhomogeneous term keeping

Pu = pu + (e/c)Au = canonical momentum that is U(1) invariant

where kinetic momentum SR 4-vector

pu = (h/i)&/&x^u

operates on charged source matter fields Psi

F = electromagnetic field 2-form, i.e. curvature in U1(x) fiber space

dF = 0

d*F = *J

d*J = 0

are the EM field equations and local current density conservation laws with EM field action density 0-form

~ *(F/\*F)/4 in 3D + 1

1916 GR Gravity is a UNIVERSAL local spacetime symmetry T4(x) for ALL matter field actions.

ONE EM F 2-form is replaced by TWO 2-forms

curvature R^a^b = dS^a^b + S^ac/\S^cb

torsion T^a = de^a + S^ac/\e^c

forced to zero as adhoc constraint.

Where spin-connection S^a^b is like EM field's A.

S^a^b is not a 6-parameter disclination Lorentz group tensor. Here Lorentz group is like U(1)

However, when you map S^a^b to Levi-Civita {^uvw} - complicated formula in Rovelli Ch 2 then Levi-Civita is like EM A with respect to local T4(x) group.

GR is a lot more complicated.

However

e^a = I^a + @A^a = e^audx^u

ea = Ia + @Aa = e^ua(&/&xu)

is like EM's canonical momentum

P = p + (e/c)A

i.e. more precisely

the local T4(x) covariant partial derivatives on all matter fields are

D^u = ea^uP^a = (I^ua + @A^ua)P^a

P^a is matrix rep of RIGID T4 of the source fields Psi

e.g. 2x2 matrix for Weyl spinor source fields etc.

4x4 for Dirac spinors

the localization is in the "phases" A^ua(x).

This universality is the REAL equivalence principle as minimal T4(x) gauge coupling of tetrads to matter fields.

The U(1) gauge transformations have no direct physical meaning, but the T4(x) do - they connect coincident LIFs with LNIFs!