I think what's left unsaid in the paper is that if a Lorentz-invariant speed exists, then it is unique. (This is easy enough to show.) Therefore, once you show

I think what's left unsaid in the paper is that if a Lorentz-invariant speed exists, then it is unique. (This is easy enough to show.) Therefore, once you show that the speed of light in a vacuum is Lorentz-invariant, then it has to be the unique Lorentz-invariant speed (i.e., the universal speed limit).