> Talking about what a photon "sees" is a non-starter

In QED or other photon-containing QFTs like the Standard Model, it is not only a "starter", it is essentially the point of the thing. Surely the photon "sees" the electron in a Compton scattering?

Photons obey the spin statistics for massless gauge bosons, so they can pile up at one point in spacetime. Equivalently, null geodesics can intersect. There can be a spacetime-point occupied by a pair of photons while the rest of spacetime is filled with points where they are occupied by only one (or neither).

(Indeed, the Penrose theorem arises because null geodesics, and photons travelling on them, can collide and stay collided, so for this and other reasons there may be multiple points in spacetime occupied by both of our pair of photons.)

One must be able to understand that for a photon there may be an early/late distinction -- either it is in a gravitational singularity or it isn't; either it is occupying a point with the other of our pair of photons or it isn't; either it's scattering or isn't. Following the photon and thinking about what it experiences, we can't use proper time, since it will be identical (practically[1]) everywhere along its path; likewise, spacetime intervals for all photons travelling on null geodesics are zero. The need for this understanding increases with photon interactions with charged matter: a pair of photons from different quasars arriving at a pair of detectors here tells us something about the location in spacetime of the quasar and about the metric expansion of space. Using cosmological coordinates, for each of the pair there is a _time_ before which no point in space contains it, and a time when they arrive at a detector on or around Earth. Given that each of the pair is from a different quasar, generally the emission time will be different. Moreover, at emission the wavelength of the photon will be shorter, and it will be longer at later times, right up to detection.

Proper time (and proper length for wavelength) makes a mess of this in general, but we do not need to parameterize a non-timelike geodesic with proper time. Instead, for a null geodeseic, we use an affine parameter and consider its properties at each value of the affine parameter.

For each of our quasar photons, at different affine parameter values the photon has a different energy-momentum. For the Minkowski vacuum photons in the first paragraph above, for a given affine parameter value, one photon is in vacuum or it is at the same point as the other photon. For accuracy, when we are dealing with physical photons, the stress-energy-momentum tensor must encode this. As we get away from simple Minkowski vacuum, we want all the tensor-fields to be accurate.

We can recast your second paragraph's second sentence as taking the limit of the geodesic in which it becomes null, parameterizing by an un-rescaled extremized proper time. We can simply rescale so that we always have a different value at each point on the geodesic as we take this limit. Indeed, we can substitute any monotonic function of spacetime position on the geodesic and get a way to compare field-properties. For a timelike geodesic, proper time tends to be a "good" choice. The affine parameter, the unique monotonic function that satisfies the geodesic equation, tends to be a "good" choice for a null geodesic, in particular because it preserves the tangent vectors under parallel transport.

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[1] Photons generally do not move at c in a medium, and one can describe this in a variety of equivalent ways, some of which demote a photon from a null curve to a timelike one while interacting with the bulk of the medium. There are serious proposals to study laboratory ultra-slow light in the context of General Relativity. From a theoretical perspective, Gordon in 1923 outlined an https://en.wikipedia.org/wiki/Optical_metric in which light follows different geodesics from the null geodesics sourced by the metric tensor, and there are various approaches to extending this to media with nonnegligible dispersion. In this sort of approach, a photon or a pulse of classical light can have a pretty conventional proper time.