To quote a movie: amazing,
everything you just said is wrong. Except maybe for the first line, but you are applying it to the wrong person; so I will give it partial credit.
By what you said above, it is evident to me that you lack anything being a basic grasp of differential geometry. This i assessment comes from the last two paragraphs you wrote, in particular, but the second half of the third paragraph is also telling. You also failed to appreciate that Newton-Cartan theory does not have a metric; it has a connection (Christoffel symbols/covariant derivative), but no space-time metric.
While writing my other reply, there was a much longer post that ended up being scraped because it was a bit of a rant and I thought it might not have been necessary. I now see I was wrong.
Diffeomorphism invariance is not a property only of the EFE. The entire theory, and all equations that come from it, have this property. It was called "principle of general covariance" by Einstein; and it basically boils down to a requirement that the laws of physics must be written in terms of geometric objects (scalars, vectors, tensors) so that they (the equations) will have the same form on all coordinate systems (and yes, I am aware of the distinction between active diffeomorphisms and passive diffeomorphisms; they don't matter here). This principle had come under fire since 1917 (at least) because many authors felt it had no real content - because any physical theory can be written in such a way. You can read about this on any graduate-level textbook on GR, or on lecture notes for such a course. For example, check Sean Carroll's (
https://ned.ipac.caltech.edu/level5/March01/Carroll3/Carroll5.html).
Now comes the part were it is a bit of a rant: usual formulations of Newtonian mechanics and SR are not written correctly: they are written assuming a certain class of coordinates, inertial frames, and lack things that would make them lose diffeomorphism invariance. Specifically, they do not make use of covariant derivatives, and they do not use a general metric. This additionally includes not making distinctions between vectors and 1-forms, and etc, in Newtonian mechanics.
When you correct these deficiencies, you get versions of Newtonian mechanics and SR whose laws are the same on all coordinate systems. Newton-Cartan theory is one of these.
The laws for corrected SR match the versions used by GR, by the way, except that the metric is not a solution of the EFE, but is ultimately obtained by doing coordinate transformations from a Minkowski metric. The usual rules for "lifting" laws from SR to GR (partial derivatives to covariant derivatives, etc) are not needed with a proper formulation of SR because they should have been baked into SR instead.