The mathematics are sound, but the reasoning around is unclear to me. The derivation shows that Lorentz transformations and Galilean transformations are the only ones that allow for the equivalence of all inertial frames, which is a nice result. But it clearly does require the additional assumption of an invariant speed to conclude that Lorentz transformations are anything more than a mathematical curiosity.
So what have we really gained? Since we still need the extra assumption that an invariant speed actually exists, we could've just gone the other way and done the light clock calculation to get the Lorentz transformation instead.
So what have we really gained? Since we still need the extra assumption that an invariant speed actually exists, we could've just gone the other way and done the light clock calculation to get the Lorentz transformation instead.