Why do we know that light must follow a geodesic? THE CONTEXT:
Some context to my question: Einstein once posed the thought experiment of a man falling inside a closed box. Just before the box was dropped, a photon was fired horizontally moving from the man's left.towards his right..By the equivalence principle, the man has no.way to realise he is falling in a gravitational field..To him, it is as if he is in zero gravity. But that would mean that he must "see" the photon continue to move from his left to his right, HORIZONTALLY. But then consider another man outside the box who is not falling. To him the same photon must follow a curve.
This is where the geodesic question comes in. If you know that light.must  always follow the geodesic, and that light is following a curve, then that automatically implies that space must be curved, since a geodesic in flat space is a straight  line. But, if there was no evidence that light followed a geodesic; then it could be thought that space is flat, but light is simply taking a curved path in flat space.  
MY QUESTION:
So my question is this: If we put general relativity itself to one side, what evidence tells us that light must travel along a geodesic? Is it possibly the principle of Snell Law that the optical path is a stationary point?
 A: You are saying that it could be that spacetime was flat, and light followed a curved path.
Let me flip your question around. How could you differentiate, these two cases:


*

*all known particles are following a bent path when they interact with the gravitational field (in your case spacetime is flat, so spacetime is flat, but particles follow bent paths), meaning that gravity does not curve spacetime itself, just curves the paths of particles

*spacetime itself is bent (because gravity bends it), so everything we know, all particles must follow its curvature (since they exist in curved spacetime, they must follow it)
Experimentally, you could not tell.
We use the phrase gravity curves spacetime, and it is confusing, I understand.
You can take electromagnetism. The only reason we do not use the phrase the EM field curves spacetime, is because there are particles we know, that are not affected by it, so the EM field does not bend their paths (and because they can bend in two different directions). So the EM field cannot curve spacetime. Or in your case, the EM field curves spacetime, but some particles (EM neutral) follow a different path, not the curved path. This curvature of EM charged particles is because of the interaction between the particle and the EM field. Not because the EM field would curve spacetime.
In reality, in the case of gravity, we cannot tell. It could be either way. The most commonly accepted view is that gravity bends spacetime, and the curvature itself is the deformation of spacetime itself. All known particles (not just photons) follow this geodesic.
But this is not correct. All known particles with stress-energy will follow this geodesic. Do we know of any particles in the SM that do not have stress-energy? No. So all particles in the SM follow these geodesics.

The geodesic is simply the path followed if no external forces act on the object. It's the general relativistic equivalent of moving in a straight line.

https://physics.stackexchange.com/a/92256/132371
This says too, that no forces act on the object (photon in your case) itself, but the force (of gravity) acts on the fabric of spacetime itself, curves it, and the object (photon) will just follow the geodesic.
We use experiments to justify our models. In our experiments, we can tell whether particles follow certain paths, but we cannot tell in our experiment whether the fabric of spacetime is deformed or not. The experiments only work with particles (photons in your case). The experiments tell that GR is right, and all known particles must follow these geodesics (if no external forces act on them).
A: Fermat's pronciple states that a ray of light takes the path traversed in the least time (or more precisely, in stationary time). This is a consequence of the principle of stationary action.
In a free metric spacetime, a geodesic defines the shortest interval between two events. The geodesic equation is obtained using the principle of stationary action with the metric (or equivalently its square root) as the Lagrangian. This implies a movement in empty space with no external forces.
By varying this action we obtain the Euler-Lagrange equations for geodesics in the given spacetime. Depending on the type of the interval (timelike, lightlike, or spacelike), we get three corresponding types of geodesics.
So yes, Fermat's principle in a free metric spacetime is equivalent to the fact that light travels along null geodesics.
