I am always looking for geometric interpretations, and as I understand it, the Einstein field equation basically says 3 things: (1) spacetime is a manifold, (2) free particles follow geodesics in the manifold, and (3) the convergence of geodesics along a given direction in the manifold is proportional to the amount of mass/energy flowing in that direction. #3 is the field equation, #2 the equation of motion.
Which is why mass/energy can be identified as the convergence of geodesics.
With EM, I guess you could say the problem is that there's no free particle. Only charged particles respond to the EM field in the first place. As far as I can tell, Maxwell's equations in curved spacetime say roughly that a charge's 4-acceleration is orthogonal to its 4-velocity and lies in the plane containing the 4-current that produced the field. Even if that's accurate, it's not simple.
But what if charge is just a more sophisticated interplay of geodesics? Something analogous to a helix, where the orientation of the helix about its time axis determines the sign of the charge, and perhaps the helix expands outward to generate the EM field, and when two helices meet they can merge if they are locally spinning the same direction, thus causing attraction, and vice versa for repulsion. That's too simplistic, but there are many structures that can exist in a 4D manifold and why couldn't they account for EM behavior?
I've read that Kaluza Klein theory combines GR & EM, but that's 5D, and it's pretty compelling to think the universe could be constructed simply from the 3D space and 1D time we know so well.