So Einstein's theory of special relativity as far as I know is essentially axiomatized, saying "Assuming the speed of light $c$ is constant in ALL frames of reference, what happens?" We know now of course that special relativity has a load of evidence behind it and quite clearly applies very well to many situations, but at the time period physicists didn't have that evidence that the speed of light was ALWAYS constant. It seemed it, but then again all of the "moving a fraction of the speed of light" analogies we use to explain special relativity had obviously never been seen experimentally. If physicists had no reason to believe the speed of light was constant to that level of consistency, why was special relativity accepted so widely and quickly? Or did they in fact have physical phenomenon which special relativity and only special relativity seemed to explain?

• Einstein set out to show what the consequences would be if Maxwell's theory were true. That's a goal that he could have achieved regardless of whether or not Maxwell's theory actually was true and, regardless of whether or not he believed it to be true. – Solomon Slow Jul 10 '17 at 17:10

The seeds of special relativity were sown when it was shown the in vacuo solutions of Maxwell's equations are not Galilean-invariant and predict their electromagnetic waves have speed $\dfrac{1}{\sqrt{\mu_0\varepsilon_0}}$. This speed matched experimental light-speed measurements, suggesting light is electromagnetic radiation. Most physicists at the time thought the solution was that Maxwell's equations are valid in a privileged reference frame with respect to which Earth has a non-relative velocity. However, this implied light-speed measurements could vary with the seasons, because of Earth's location in its orbit. Einstein realised the true was that physics needed to be Lorentz-invariant like Maxwell's equations, so there was no privileged reference frame. Einstein therefore could have phrased one of his axioms as, "Electromagnetic waves in vacuo are of speed $\dfrac{1}{\sqrt{\mu_0\varepsilon_0}}$ in all directions in all reference frames". For whatever reason, he happened to talk about light instead, since by then its electromagnetic status wasn't really disputed. However, it doesn't matter too much which thing is posited to behave so in a vacuum; as of general relativity we could even use gravitational waves instead.