Why is the photon clock equivalent to all clocks?

Invoke the principle of relativity.
An inertial observer carries both a light clock and a mechanical wristwatch, which agree when all are at rest. If they don't agree when the inertial observer is moving [with nonzero constant velocity] carrying these clocks, then that observer can distinguish being at rest from traveling with nonzero constant velocity.

UPDATE:
Q: What makes the photon clock special among all other clocks?
A: Simplicity.
It's easier to formulate, analyze, and interpret than other clocks.
If the principle of relativity holds, it must turn out that one can eventually analyze any clock and get the same result as the light-clock---it probably takes a lot more analysis and interpretation [of the device, the surroundings, and the interactions].

Based on my current understanding of the topic the light clock is not a proof of time dilation but simply a clue, pointing at it. You are absolutely right in stating that the light clock is not a proof. In fact there isn't any proof of Lorentz transformation at all. Lorentz transformations are not proven, Lorentz transformations are postulated. This transformations are our best guess of how time and space works in absence of gravity and acceleration. Sure we can see that our experiments, mental or physical, agree with them, but this is not a proof, is reasoning by induction at best.

This kind of mental experiments, such as the light clock, help us to guess the correct form of the transformation, but there is no way to prove them. This is a recurrent theme in physic.

Why is the photon clock equivalent to all clocks?

Stipulate that, in some inertial reference frame, there is a photon clock and some mechanical clock that are co-located and at rest in this frame.

Further stipulate that, in this frame, the clocks run at the same rate, i.e., both clocks 'tick' simultaneously.

Now, because the two clocks are co-located, all inertial observers in relative motion to the clocks agree that the ticks are simultaneous. Whatever time dilation is observed by the relatively moving inertial observers affects both clocks identically.