> The path of a charged particle in the EM field could be described as that particle experiencing a different space-time geometry
No, it can't, because the EM interaction does not obey the equivalence principle, as gravity does. The geometric interpretation of gravity relies on the equivalence principle.
To state this another way: if I put two objects with different masses at a given point in spacetime and give them both the same initial velocity in the same direction, their paths through spacetime under gravity will be the same. But if I put two objects with different charges at a given point in the same electromagnetic field and give them both the same initial velocity in the same direction, their paths through spacetime will not be the same. And this remains true even if I add "extra dimensions" to "spacetime" along the lines of Kaluza-Klein theory, to represent the EM field.
No, it can't, because the EM interaction does not obey the equivalence principle, as gravity does. The geometric interpretation of gravity relies on the equivalence principle.
To state this another way: if I put two objects with different masses at a given point in spacetime and give them both the same initial velocity in the same direction, their paths through spacetime under gravity will be the same. But if I put two objects with different charges at a given point in the same electromagnetic field and give them both the same initial velocity in the same direction, their paths through spacetime will not be the same. And this remains true even if I add "extra dimensions" to "spacetime" along the lines of Kaluza-Klein theory, to represent the EM field.