So if I want to construct a theory that says "the metric is Minkowski", but picks out different curves as the "straight lines"--curves which are not freely falling worldlines--then the only way I can pick out which curves these are is to use some non-local criterion. In Schild's case, the criterion is to pick the worldlines that are "at rest" with respect to observers very far away, as verified by round-trip light signals. But there's no local way to tell which worldlines those are; there's no local way to say, the worldline with this particular proper acceleration is the "straight line" in this particular local region of spacetime. Only the nonlocal measurement can tell us that.
We have the claim:
Locally, there is no way to distinguish freefall from inertial motion
To me, that claim IS the equivalence principle. If the equivalence principle is false, then that claim is false.
For example, we have another criterion for inertial motion, which is that "An inertial path is one that maximizes proper time". It's conceivable that that would give a different answer as to what is an inertial path than freefall. GR says that freefall = inertial, but that's an empirical question. You can't assume it.
I'm only saying that, if we are going to say the metric is Minkowski but have some "straight lines" that are not freely falling worldlines--which we must do in the presence of gravity--then we have to use a nonlocal criterion to pick out which worldlines are the "straight lines", because there is no local way to do it.
Isn't maximizing proper time a local criterion?