Isn’t this a bit too quick? Merely from the relativity principle (the SR sentence) and the fact that the speed of light is measured by a physical experiment, it doesn’t follow that the speed of light is constant - it could in principle depend on details of the experiment, such as motion of the light source.
Rather, I think to get SR we need the relativity principle plus certain experimental facts (such as the independence of the speed of light from the motion of the source). Without those experimental facts, the SR sentence is compatible with Galillean relativity as well. (Or even the Riemannian relativity of Greg Egan’s Orthogonal novels.)
I am paraphrasing for pedagogical reasons. Of course you also need the theory of electromagnetism (and thus all the experiments that establish electromagnetism) to work on anything relating to light. It’s always the “SR sentence” plus the rest of physics circa 1900. I think this should be automatically assumed: you don’t even have an equation of motion if all you assume is the “SR sentence”. No need to be pedantic, this is a pedagogical exposition not an axiomatic system.