As mfb says, GR is local, not action-at-a-distance. As taught in schools today, Newton isn't action at a distance either: gravity is due to a local field. I know of no alternatives to GR where gravity is not determined by a local field. The nature of this field can change in different theories, but it's always a local field. The whole point of LeSage is to provide an explanation for this local field in terms of unseen particles. This throws the baby out with the bathwater.
There is no theory of gravity that I am aware of that proposes this.
That logic is vacuous, in that it can be applied to any objection in any theory. "Don't worry about it. Maybe someone can fix it" is just about the weakest counter-argument I can imagine.