However, this principle is more involved in motivating GR, than functioning as a strict consequence, given the equations of GR.
So a different question would be, could a theory of varying alpha be consistent with GR? I am not sure that is so difficult. One could imagine some field permeating the universe, following a covariant law, that influences alpha.
In any case, I think GR would not be the only major modern theory affected by varying alpha. I think it makes the article more exciting to specifically mention GR.