Why is the photon clock equivalent to all clocks? I can understand why, if the speed of light is invariant, a photon clock would tick slower. I find this explanation very useful in terms of introducing the idea of time dilation (also because it allows for the Lorentz formula to be derived intuitively, only using Pythagora's Theorem).
But this approach has one important missing concept. A student might say; Okay I get why the photon clock would tick slower, but why is it an intrinsic property of time itself? Why is this not some effect of the mechanics of this specific clock? How are a pendulum clock, an atomic clock, circadian rhythms, a chemical clock, etc... all equivalent to the photon clock? Why the slowdown of the ticking of the photon clock is a probe on the very nature of all clocks and time itself and not just a probe on the nature of this particular clock (more so if we consider that the explanation relies on the specific mechanism of this clock to work)?
For example some students might reason; a pendulum clock would slow down on the lunar surface, since the gravity is lower and therefore the pendulum would have a larger period, but we don't immediately jump to the conclusion that time itself has slowed down on the Moon with respect to Earth (in fact, ironically, in general relativity it is the other way around), just that the technical features of this particular clock make this happen because we have altered its functionality by altering the physical enviroment where it operates. The same could be said of a spring clock submerged in water for example. But if we don't think that the Moon gravity slows time with respect to Earth's just because the pendulum clock ticks slower, or that water slows time just because the spring clock ticks slower, then why should we think that moving at a certain relative speed slows the flow of time just because the photon clock ticks slower?
 A: 
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.
A: 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.
A: 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].
A: A mechanical clock is made of cogs and springs which transfer forces to the clock hands along complicated pathways. But if you look really close, all of those cogs and springs are made from protons and electrons, held together by electromagnetic forces and they push on each other by exchanging photons, just like in the simplified photon clock model.
A: In my answer here, you can replace the horizontal photon clock with an atomic clock, a pendulum clock, or the clock of your choice.
A: You are asking "Why the slowdown of the ticking of the photon clock is a probe on the very nature of all clocks and time itself and not just a probe on the nature of this particular clock (more so if we consider that the explanation relies on the specific mechanism of this clock to work)?"
Now it is because the photon clock is the fundamental clock, it is like the "elementary particle" of clocks. I will use here the theoretical explanation of the photon clock where there is always vacuum inbetween the mirrors, that is, the photon bounces in vacuum always.
This is because the photon clock only (disregarding the mass of the mirrors) uses one fundamental underlying element of the universe we live in, that is light (EM wave) and the speed of light. All other clocks are more complex to explain, yes, even the atomic clock.
The speed of light is c always, when measured locally in vacuum. Furthermore, all inertial observers will agree on this one speed, independent of their relative motion.
You are saying, that in the case of other clocks "Why is this not some effect of the mechanics of this specific clock?" and "just that the technical features of this particular clock make this happen because we have altered its functionality by altering the physical enviroment where it operates.", now this is the crucial point in your question.
You cannot alter the functionality of the photon clock because of its simplicity, its fundamentality, its elementarity, and because it uses only one single ingredient, the speed of a single photon bouncing between two mirrors (where we disregard the mass of the mirrors).
A: The "moving" photon clock is in a frame of reference that has (in principle) a set of synchronized clocks distributed along the axis of relative motion to the "stationary" frame. So whatever is deduced about the photon clock must apply to the other clocks in that frame as they are synchronized. The fact that some other observer is measuring his clock in no way affects his clock nor its synchronization with respect to all the other clocks.
A: Your starting position is not accurate. Relativity depends on one axiom.

*

*The general principle of relativity: Local laws of physics are the same irrespective of the reference matter which a particular observer uses to quantify them.

It follows immediately that all clocks measure exactly the same unit of time (to the accuracy of the clock mechanism). They run at precisely one second per second. Time dilation does not affect the rate at which a clock measures time. It is an apparent effect, due to the way in which an observer views a moving clock. You can compare this to the observed length of a rod held at an angle to the line of sight of the observer. The observer does not see the true length.
Unfortunately, early on in relativity, Einstein thought that the observed properties (length and time) should be thought of as the physical properties. Later, he changed this, and introduced the general principle which makes clear that the physical properties of an object are measured by an observer moving with the object. Even more unfortunately, over 100 years later, low grade and popular accounts have not caught up.
A: It is a common misconception that time dilation is some effect that impedes the action of clocks as a result of their motion, which then leads people to ask how come different types of clocks are affected in the same way.
Time dilation refers to the fact that the time interval between two events in one frame can be less than the time interval between the same two events in another. It is the actual time interval that varies, not the behaviour of clocks. If the time interval is 4s in one frame and 3s in another, it means that the interval in the second frame is actually a second less, and any accurate clock will measure it as such.
If you flash a light that is detected some time later in your frame, and I am passing by, in my frame the light will have travelled a different distance than in yours, so if the speed of light is the same for both of us, we must therefore disagree about the duration of the light's journey- ie, the time interval between the flash and the detection varies by frame, and any type of clock will measure that.
A: It is because all clocks are derivatives of the photon clock. In other words, all functional clocks could be reduced to a photon clock, but not necessarily vice versa (in practice, actually never). It all stems from the general theory of relativity by Alberet Einensten and the fact that photons are the mediating molecules of electromagnetic interactions.
