tl;dr Neighbouring points on a null worldline are not instantaneous because c is not infinite; distant points on the same worldline are even less instantaneous. There are many useful ways of capturing the "distant" in "distant points".
An element of light traces out a worldline. On any worldline we can apply whatever labelling-of-points we want since relativity is a coordinate-independent theory. We can label the points of a worldline with greek letters, hieroglyphs, roman numerals, natural numbers, real numbers, whatever we like and in any order we like.
One can build an infinity of calculationally-useless or misleading sets of coordinates on these worldlines for things heavier/slower than light. But one can also build an infinity of calculationally-useful and non-misleading coordinates for them, and many of those make use of the invariant spacetime interval. The same applies to coordinates for massless things / things that move at the speed of light, even though the invariant spacetime interval for light in free-fall is always 0, even if it is in free-fall for billions of years (like light from distant quasars, or the cosmic microwave background).
A calculationally-useful ordering applies a monotonically increasing order from the start to the end of a worldline in a time-orientable manifold (our universe is time orientable: smaller and denser in the past, bigger and sparser in the future). For timelike worldlines (i.e., anything that is always slower than light), almost always the most useful ordering is proper time.
But we cannot calculate proper time on a null (lightlike) worldline, so we will want some other monotonically increasing ordering function on the worldline, and ideally one with which we can solve the geodesic equation. Such a family of orderings is not only known, but has been textbook material since 1970 (Spivak's introduction to differential geometry). It's the affine parametrization.
For lightlike observers there is thus a useful and well-defined notion of time: the affine (parameter) time. This is different from but analogous to the proper time available to timelike observers. We can do standard vector physics on a photon using affine time, e.g. we can calculate its phase at various points along its trip from point A to point B. (Indeed, talking about a photon's quantum wavefunction, the affine parameter is proportional to its phase). We can also take the derivative of position with respect to affine time as a momentum that accurately captures the gravitational redshift or blueshift between two points on the null worldline.
An element of light traces out a worldline. On any worldline we can apply whatever labelling-of-points we want since relativity is a coordinate-independent theory. We can label the points of a worldline with greek letters, hieroglyphs, roman numerals, natural numbers, real numbers, whatever we like and in any order we like.
One can build an infinity of calculationally-useless or misleading sets of coordinates on these worldlines for things heavier/slower than light. But one can also build an infinity of calculationally-useful and non-misleading coordinates for them, and many of those make use of the invariant spacetime interval. The same applies to coordinates for massless things / things that move at the speed of light, even though the invariant spacetime interval for light in free-fall is always 0, even if it is in free-fall for billions of years (like light from distant quasars, or the cosmic microwave background).
A calculationally-useful ordering applies a monotonically increasing order from the start to the end of a worldline in a time-orientable manifold (our universe is time orientable: smaller and denser in the past, bigger and sparser in the future). For timelike worldlines (i.e., anything that is always slower than light), almost always the most useful ordering is proper time.
But we cannot calculate proper time on a null (lightlike) worldline, so we will want some other monotonically increasing ordering function on the worldline, and ideally one with which we can solve the geodesic equation. Such a family of orderings is not only known, but has been textbook material since 1970 (Spivak's introduction to differential geometry). It's the affine parametrization.
For lightlike observers there is thus a useful and well-defined notion of time: the affine (parameter) time. This is different from but analogous to the proper time available to timelike observers. We can do standard vector physics on a photon using affine time, e.g. we can calculate its phase at various points along its trip from point A to point B. (Indeed, talking about a photon's quantum wavefunction, the affine parameter is proportional to its phase). We can also take the derivative of position with respect to affine time as a momentum that accurately captures the gravitational redshift or blueshift between two points on the null worldline.