But isn't that use of the speed of light similar to (and in some sense inferior to, depending on what you are studying) the vibration of an atom as is currently used as our most reliable frame of reference for measuring 'relative' movement, aka measuring 'time'?
This is actually, probably the most common question for special relativity. Many people would ask, why if a clock of light slows down, then everything must slow down too? The problem lies with Maxwell's equations, which imply that the speed of light should be a constant. And that cause problems with Galilean invariance. As if the speed of light is c only in a static frame, then you can measure the speed of light in your reference frame and you can tell that you are actually moving at a certain constant speed without looking outside. That'd be very bad.
Since Maxwell's equations are very accurate descriptions of electromagnetic waves, and Galilean invariance is an important basic law of physics, obviously one needs to find a way to incorporate both without contradiction. The first attempt was the "ether," which suggested that ether is an universal medium for the propagation of light. Then the speed of light is only constant w.r.t. the ether, not everything else. So we have the famous Michelson-Morley experiment, which tried to verify that light travels at different speed for different direction (as the earth is rotating around the sun, and the sun is rotating around the center of the Milky Way, which should be a quite complex movement w.r.t. the "static" ether). However, nothing was found. There is no difference between the speeds of light for different directions. So there must be other explanation. Lorentz suggested a transformation which makes Maxwell's equations compatible to Galilean invariance, but he didn't provide a satisfying explanation for the transformation (which is simply a mathematical construct). Einstein's special relativity provides the satisfying explanation for Lorentz transformation.
The whole point of everything is relativity. If I have a circle and let light travel that circle in one direction, light will have passed that circle at the speed of light, and the speed at which the circle has filled is distance * c. But now if I let light travel that circle in both directions, light will have passed that circle in (distance/2) * c. Therefore, if I were to assess the speed at which light travels left through the circle from the light travelling right through the circle, the speed at which one distances itself from the other (or closes in on the other, as this is a circle) is twice the speed of light.
The problem is, first, this is not compatible with experiment results. No experiment detects different speed of light (in the vaccum), whether the light source is moving at high speed or not. Second, if Maxwell's equations are correct (which is very likely to be the case), if you observe "twice the speed of light" then you can violate the Galilean invariance, that is, you can determine whether you are moving at a constant speed or not. This is also very unlikely.
Now, if there's another very nice clock which allows you to observe difference between it and a clock of light (e.g. it does not slow down when moving at a high speed), then you can use this clock to violate Galilean invariance. Since we believe that Galilean invariance is correct, there must not be such clock. That is, everything must slows down as the clock of light.
, the that's relativity for you.
