Let's see if we can expand your understanding a little, if you like. I'll also try to make some sense in case someone else sees this in the future.

The tl;dr (cf final paragraph below) is that even though it is hard to make sense of a photon-with-wristwatch (does it ever tick?), General Relativity is a theory in which we can calculate the behaviours of an object travelling at the speed of light interacting with other things travelling at the same speed or slower. Moreover, in General Relativity light can redshift and blueshift due to local gravitational influences and by travelling cosmological distances through the metric expansion of space. We may need to know how much redder it is at some point in spacetime compared to some other point in spacetime, and to some extent that question depends on the photon and its history. We can make sense of the history part as follows:

First think four-dimensionally. Rather thinking of the propagation of than an object with some spatial extent from point A in space to point B in space at a later time, let's start with a pointlike object, literally of dimension zero.

Let's promote this 0-d object in 3-d space to a 1-d object in 4-d spacetime. That promoted object is a worldline. Points A and B are now simply two different points on the worldline.

Now we need a distance function along the worldline. As we're interested in the distance between A and B we want to choose any arbitrary function that gives a monotonic value along the worldline, for example starting with 0 at A and ending at some larger number at B, with every point in between having a value between 0 and the value at B.

If our worldline is everywhere timelike then we can simply extract the proper time at each point on the worldline from the line element of the spacetime's metric.

However, for a photon, the worldline is everywhere lightlike, and the proper time is undefined at every point, so we can't use it. Attempting to reason about this, by for example setting the proper time to the same value (e.g. zero) everywhere on the worldline, leads to incorrect conclusions in your explanation a couple postings back.

Instead, we can choose an arbitrary function. The requirement, again, is that the value at A is less than the value of B and every point inbetween along the worldline. We can without difficulty do better than using a function that returns an undefined or identical value at A, B and points inbetween.

Ideally we can choose a function that gives a number at every point on the worldline, even extending beyond A or B, and which we can relate to the the curvature of the spacetime in which we find our worldline. If the interesting part of our worldline is always on a single null geodesic, then there is a good choice of function: the affine parameter. It satisfies the geodesic equation, and gives the right tangent vector for anywhere along the geodesic.

Decomposing back into our 0-d particle, this means that at any point in anyone's time, one can use the affine paramater on the 0-d particle's trajectory to figure out how its vectors parallel evolve between two points on that trajectory. Indeed, an observer from her or his perspective can predict where the particle will go, and where it has come from.

An object with spatial extent goes from a worldline to a worldtube, but the principle is the same: each point on the worldtube can be distinguished using an appropriate function. For a photon, we still use the affine parameter, because photons travel on null geodesics. They just freely-fall through curved spacetime until they have some direct interaction; that's all being on a null geodesic means.

In a spacetime with timelike worldtubes and lightlike worldtubes we have a network of intersections in spacetime, and we are interested in the behaviours at those intersections. Anyone can apply an arbitrary system of coordinates -- or distances along each intersecting worldtube -- and calculate physical quantities at the intersection points in spacetime. Because these intersections are in spacetime there is no "I got to that point in space first and just missed it"; time really doesn't matter -- the point is that at the same point in spacetime, two worldtubes interact.

General Relativity (in our 4-d universe) almost always lets us build a small region of locally flat spacetime around such an intersection, such that we can then use the Special Relativity background for calculations. The Standard Model, Quantum Electrodynamics, and other relativistic theories describing light all work in this flat-spacetime Special Relativity bubble, even if the wider spacetime is curved.

Alternatively, we know how to foliate spacetime along well-chosen timelike axes, which also slices all worldtubes into objects of spatial extent moving from one spatial slice of spacetime to the next. This is a 3+1 formalism on General Relativity, but it's important that it's still General Relativity with worldtubes on curved spacetime (which has a metric for which we can find lightlike and timelike geodesics). But even in such approaches we don't have the behaviour photons making no sense against the chosen timelike axis. Indeed, we can look at the lapse function which generates a proper time increment even for photons: \delta\tau = \alpha(t,x)\delta t.

In physical cosmology, we can slice up our universe along a time axis called the scale factor, and then we find a lapse function (and shift vector) for everything in the spacetime. A free-streaming photon's worldtube has properties (e.g. wave-vector) that are well defined at each scale factor.

Whether a cosmic microwave background photon can ask itself whether it feels tired (redder) as we take the scale factor closer to our present day, or otherwise check a "scale factor wristwatch" or look out the window to see where the horizon is, is really a problem for metaphysics. We can calculate it in General Relativity, and work out that it is redder today than it was at the surface of last scattering. Moreover, we can calculate counterfactuals: a CMB or quasar photon that freely streams to us along a path that never takes it near a galaxy (other than ours) versus a CMB or quasar photon that takes a trajectory that brings it near a massive galaxy or cluster will have different redness on arrival here. Compare the redness to a clock-on-spaceship: along the first trajectory nowhere near massive objects, we get one reading of the clock on arrival, but along the trajectory which passes near the massive galaxy or cluster we will have a different (earlier) reading. The second clock ticked slower because it went near a massive object. The second photon is redder because it went near a massive object.