> your velocity through time approaches zero

Proper time, τ. But your proper time and my proper time are different, and are only coordinate times, monotonic labels on timelike curves. That means τ is an affine time.

There are other affine times available. And we have to choose an affine time other than τ for null geodesics, the curves which photons (in vacuum) travel along. So, instead of proper time, for photons there's the affine parameter.

The same rule applies: there is nothing special about the affine parameter. I don't have to use a photon's affine parameter when describing physics any more than I have to use the proper time of an ultrarelativistic electron. And I can convert physics done in one splitting of spacetime into space+time to a different splitting of the same spacetime into space'+time'. The coincidences in the spacetime are unchanged by the switch of how we split -- splitting is just a change of coordinates.

If we divide up spacetime into space+time where the choice of time axis along which to order spatial volumes is my proper time, the emission of a photon in the photosphere of the sun and its subsequent destruction in my eyeball happen in a definite temporal order: there is a time delay. There is still a temporal order if we divide up space+time using your proper time, or that of an ultrarelativistic electron.

There are some special technical aspects of dividing up spacetime into space+time using a photon's affine parameter as the time axis, but it's certainly doable. A photon shares a different 3d spatial volume with many other things at each affine parameter labelled point on the photon's curve. Phases of orbits, stages of an elastic collision, and so forth are among the things "snapshotted" into each 3-d spatial volume the photon finds itself in. Those evolve from one 3-d spatial volume to the next to the next.

So returning to the previous picture: if we split up space+time according to the affine parameter of a photon, there will be some 3-d volumes at which the solar photon is much closer to the sun than to my eyeball, and some 3-d volumes where it is much closer to my eyeball than to the sun. The same is true for the solar photon itself: some affine parameter values put it closer to the sun than to my eyeball, and at some affine parameter the photon's journey ends.

If a photon's trajectory is through curved spacetime, there will be a difference in momentum between two points on the trajectory (we are not restricted to the photon's creation or its destruction), which we can calculate using the affine parameter. The physical interpretation is that the photon undergoes a redshift or blueshift between two points in spacetime.

Note that I did not take the lim v->c approach you did in your first paragraph because there are geometrical differences in a Lorentzian spacetime between a null geodesic and a timelike one, even if the timelike geodesic is associated with a speed arbitrarily close to c. The photon is almost always on a null geodesic. A non-massless observer will never be on a null geodesic.

> Since photons move at c

Photons can be made to move slower than c, in which case they are not on null geodesics, and therefore proper time might be suitable for them.

Photons moving at c must be on null geodesics.

Proper time -- one particular affine time -- is undefined on null geodesics. However, they experience a different affine time. One can interconvert, so it does not make sense to say that photons "experience zero time".

Lastly, if the totality of the photon's curve through spacetime is on a null geodesic, the photon won't be able to experience much of the universe evolving around it as it flys away from its creation. However, segments of a photon's curve can be other than null geodesic motion as (for example) they cross through wispy gas clouds or strong magnetic fields in space. Temporarily slowed light <https://en.wikipedia.org/wiki/Slow_light> can in principle receive news of the world. This could have happened for a photon emitted billions of years ago from a high-redshift quasar en route to the JWST.

Extra reading:

https://en.wikipedia.org/wiki/Initial_value_formulation_(gen...

Blau, Frank, Weiss 2006 (section 3 "Brinkmann coordinates are Fermi Coordinates:", 4 "Null Fermi coordinates, general construction", 5 "Expansion of the metric in null Fermi coordinates") arxiv version https://arxiv.org/abs/hep-th/0603109 (link to publication in Class.Quant.Grav. is on the abstract page).

"What is the physical meaning of the affine parameter for null geodesic?" https://physics.stackexchange.com/questions/17509/what-is-th...

The first nuclear clock will test if fundamental constants change