Your real question is: why is the fine-structure constant the value that it is? The answer is unknown. In the end, you can find the relationship called the fine-structure constant has a particular value. The reason the speed of light (or of any massless thing) seems to be something specific is its relationship to other things. $0$ gives Galileo’s fixed absolute time, and any positive value gives special relativity with a speed limit. In the linked paper, it’s “not self-consistent”, but Greg Egan worked out this case in detail). There are 3 cases: negative did not work (in the paper I’m remembering, the math chokes. In the paper an arbitrary non-determined value $Z$ came out of it. In the end, it shows that the familiar special relativity is the inescapable answer. ※Īnother paper with the same idea (but different specific axioms) is One more derivation of the Lorentz transformation by Jean-Mark Lévy-Leblond published in 1976 (thanks bdforbs). And others that are derived from an initial version by Mermin. I can’t find that one, but it’s easy to find Nothing but Relativity.
There is a wonderful paper I remember reading which uses only basic algebra only to determine the most general form of the formula to add velocities, based only on general principles of symmetry (what works here also works there, etc.).