It's not quite clear to me what you're looking for here. It's straightforward to use the Lorentz transformation to show that if the two clocks are synchronized in the train's coordinate system (the primed system), then the rearward clock is observed to be ahead of the forward clock in the platform's coordinate system (the unprimed system).
So, why would the author of the linked document use photons to illuminate (heh) this result?
Well, one way to synchronize spatially separated clocks at rest in one's frame of reference is by the exchange of light signals. This synchronization convention is known as Einstein synchronization and it guarantees that one-way speed of light (in vacuo) measurements will yield $c$.
Honestly, I think a better approach to showing that the rearward clock is ahead in the unprimed system is to observe the train's clocks synchronization procedure from the platform.
Imagine that the photon source is position precisely at the midpoint of the train car. The photon source simultaneously emits a forward traveling photon and a rearward traveling photon.
The forward and rearward clocks are synchronized in the primed system when they record identical arrival times for their respective photons (all observers agree on the time each clock reads when their respective photon arrives).
As observed from the platform however, the elapsed time for the rearward photon to propagate from the source to the rearward clock is less (since the rearward clock closes the distance) than the elapsed time for the forward photon to propagate to the forward clock (since forward clock opens the distance).
Thus, if both clocks record identical time of arrival for their respective photons, it can only be that the rearward clock is ahead of the forward clock in the unprimed system.