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?