My question is essentially this one, but since that one never got a satisfactory answer, and it's of such fundamental concern, I'd like to try asking it in a different way.
The question is whether general relativity offers a reason why gravitational and inertial mass are equal. But this is not fully answered by the law of geodesic motion; that merely removes gravity from the list of forces, and changes the meaning of "acceleration" to "deviation from a geodesic path". So the question becomes: how does having gravitational mass cause a body to resist deviating from its own geodesic path, when a non-gravitational (eg. EM) field is present?
Meanwhile, the field equation associates "gravitational mass" with positive timelike Ricci curvature (actually, positive Ricci curvature in all spacetime directions, if that matters).
And finally, since we are working in the confines of GR, we must represent the "non-gravitational external field" as yet another region of non-zero Ricci curvature. If we're talking EM, its form would thus be given by the electromagnetic stress-energy tensor. This does not capture the full character of the EM field, but it's the best GR by itself can offer. So finally we arrive at the fully-GR version of the question:
Does the presence of higher positive Ricci curvature along a worldline imply that worldline will deviate less in the presence of a given external background of non-zero Ricci curvature?