When has this ever worked?
Some people think that Einstein created special relativity this way. That is completely false, and special relativity was developed directly from experiment.
Nevertheless, it seems possible that special relativity could have been derived from first principles. Here is a recent paper that gives such a derivation:
An exposition of special relativity without appeal to "constancy of speed of light" hypothesesI have made similar arguments on this blog, as well as taking it further to electromagnetism and the standard model of particles.
We present the theory of special relativity here through the lens of differential geometry. In particular, we explicitly avoid any reference to hypotheses of the form "The laws of physics take the same form in all inertial reference frames" and "The speed of light is constant in all inertial reference frames", or to any other electrodynamic phenomenon. For the author, the clearest understanding of relativity comes about when developing the theory out of just the primitive concept of time (which is also a concept inherent in any standard exposition) and the basic tenets of differential geometry.